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Comparison of structural analysis methods for masonry arches

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Title:
Comparison of structural analysis methods for masonry arches
Creator:
El Tomi, Asrar Ahmed ( author )
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University of Colorado Denver
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English
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Thesis/Dissertation Information

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Master's ( Master of science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Civil Engineering, CU Denver
Degree Disciplines:
Civil engineering
Committee Chair:
Rens, Kevin
Committee Members:
Futz, Frederick
Nogueira, Carnot

Subjects

Subjects / Keywords:
Arches ( lcsh )
Masonry ( lcsh )
Arches ( fast )
Masonry ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Abstract:
Different methods for the structural analysis for a masonry arch bridge are examined under similar conditions and the results of these analyses are compared. The MEXE method, often used in the United Kingdom, is compared to the method utilized by Ring, a software specifically intended for masonry arch bridges. The Ring method indicates approximately double the structural capacity as that from the MEXE method. Ring is also compared to the results of analyses from the Risa 2D Frame and Risa 2D Finite Element methods. The methods give different results, with Ring and MEXE method resulting in an allowable axle load. However, Risa 2D Frame and Risa 2D Finite Element lead to stresses, so are indirectly related to MEXE and Ring. The Frame and Finite Element methods proved less satisfactory than MEXE or Ring because of analytical difficulties in modelling the non-linear response of joint separations between individual stone units within the arch.
Bibliography:
Includes bibliographical references.
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System requirements: Adobe Reader.
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n3p
Statement of Responsibility:
by Asrar Ahmed El Tomi.

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University of Colorado Denver
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on1015342677
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Full Text
COMPARISON OF STRUCTURAL ANALYSIS METHODS FOR
MASONRY ARCHES
by
ASRAR AHMED EL TOMI B.S., ETniversity of Tripoli, 2007
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering Program
2017


This thesis for the Master of Science degree by Asrar Ahmed El Tomi has been approved for the Civil Engineering Program by
Kevin Rens, Chair Frederick Rutz, Advisor Carnot Nogueira
Date May 13, 2017
11


El Tomi, Asrar Ahmed (M.S., Civil Engineering Program)
Comparison of Structural Analysis Methods for Masonry Arches Thesis directed by Associate Professor Frederick Rutz
ABSTRACT
Different methods for the structural analysis for a masonry arch bridge are examined under similar conditions and the results of these analyses are compared. The MEXE method, often used in the United Kingdom, is compared to the method utilized by Ring, a software specifically intended for masonry arch bridges. The Ring method indicates approximately double the structural capacity as that from the MEXE method. Ring is also compared to the results of analyses from the Risa 2D Frame and Risa 2D Finite Element methods. The methods give different results, with Ring and MEXE method resulting in an allowable axle load. However, Risa 2D Frame and Risa 2D Finite Element lead to stresses, so are indirectly related to MEXE and Ring. The Frame and Finite Element methods proved less satisfactory than MEXE or Ring because of analytical difficulties in modelling the non-linear response of joint separations between individual stone units within the arch.
The form and content of this abstract are approved. I recommend its publication.
Approved: Frederick Rutz
m


ACKNOWLEDGEMENTS
My sincerest thanks to my advisor, Dr. Frederick Rutz for guidance, support, and encouragement for two semesters. I would never have been able to finish my thesis without his guidance. Also, thank you Dr. Frederick Rutz for preparing me to write this thesis.
My sincerest thanks to Dr. Kevin Rens for approving me to be a part of the Ely Stone Bridge team. Also, thank you to Dr. Kevin Rens and Dr. Carnot Nogueira for being a part of my thesis committee.
I would like to express the deepest appreciation to my husband, my little son, family, and friends.
Finally, thank you to Department of Engineering in the University of Colorado Denver and all the Structural Faculty in giving me the opportunity to pursue a graduate degree.
IV


TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION...............................................................1
1.1 Introduction.............................................................1
1.2 Research Objective.......................................................1
1.3 Outline of This Thesis...................................................1
II. LITERATURE REVIEW........................................................3
2.1 Introduction.............................................................3
2.2 History of Stone Masonry Arch Bridge.....................................3
2.3 Arches...................................................................4
2.3.1 Arches classified....................................................6
2.3.2 Behavior of the arch under load......................................8
2.4 Description of the Stone Masonry Arch Bridge.............................8
2.4.1 Materials............................................................9
2.5 Structural Analysis of Old Masonry Arch Bridge...........................9
III. THE ELY STONE BRIDGE......................................................11
3.1 Introduction............................................................11
3.2 Background and Description..............................................11
IV. ANALYSIS METHODS: GENERAL..............................................15
4.1 Introduction............................................................15
4.2 Methods Used for Modeling...............................................15
4.2.1 Three-Hinged Arch method............................................16
4.2.2 The Modified MEXE method............................................17
4.2.3 RING 3.2............................................................18
4.2.4 RISA................................................................19
4.2.5 Three-dimensional FEM...............................................19
V. LOADS....................................................................20
5.1 Introduction............................................................20
5.2 Design Codes and Regulations............................................20
5.3 Loads..................................................................20
5.3.1 Dead load...........................................................21
v


5.3.2 Live load “truck load”..............................................22
5.4 Load Combinations.......................................................24
5.4.1 Combined dead load and live loads...................................24
VI. ANALYSIS AND COMPARSION OF THE RESULTS................................29
6.1 Linearly Elastic Analysis...............................................29
6.2 Analysis Using MEXE.....................................................34
6.2.1 Calculate the provisional axle load.................................36
6.2.1.1 Calculate modification factors..................................38
6.2.2 Modified axle load..................................................42
6.3 Analysis Using Ring.....................................................43
6.4 Analysis Using Risa 2D Frame............................................47
6.4.1 Loads...............................................................48
6.4.1.1 Load combination with truck in position 1.......................52
6.4.1.2 Load combination with truck in position 2.......................55
6.4.1.3 Load combination with truck in position 3.......................58
6.4.1.4 Load combination with truck in position 4.......................61
6.4.2 Risa 2D Frame with three hinges.....................................64
6.4.2.1 Load combination with truck in position 1.......................65
6.4.2.2 Load combination with truck in position 2.......................68
6.4.2.3 Load combination with truck in position 3.......................70
6.4.2.4 Load combination with truck in position 4.......................73
6.5 Analysis Using Risa 2D Finite Element...................................75
6.5.1 Loads...............................................................76
VII. DISCUSSION..............................................................83
7.1 Introduction to Comparison of Different Calculation Methods.............83
7.1.1 The MEXE and Ring...................................................83
7.1.2 Risa 2D Frame and Ring..............................................84
7.2 Conclusion..............................................................89
7.3 Recommendations for Further Research....................................90
REFERENCES......................................................................91
APPENDIX
A. Field Visitation........................................................93
vi


B. Spreadsheet for MEXE Methods..............................................104
C. Report from RING 3.2 Analysis.............................................106
vii


LIST OF TABLES
TABLE
6.1. Barrel factor (The highways agency, 2001).................................40
6.2. Fill factor (The highways agency, 2001)...................................40
6.3. Width factor (The highways agency, 2001)..................................41
6.4. Depth factor (The highways agency, 2001)..................................41
6.5. Mortar factor (The highways agency, 2001).................................42
6.6. Parameters used in Ring for the Ely Stone Bridge.................................43
6.7. Load factor used in Ring for the Ely Stone Bridge................................43
6.8. Parameters used in Risa 2D Frame for the Ely Stone Bridge.................48
6.9. Dead load values in Risa 2D Frame................................................48
6.10. Total stress due to load combination with truck in position 1...................52
6.11. Total stress due to load combination with truck in position 2...................55
6.12. Total stress due to load combination with truck in position 3...................58
6.13. Final results due to load combination with truck in position 4...........61
6.14. Total stress due to load combination with truck in position 1 with three hinges.65
6.15. Total stress due to load combination with truck in position 2 with three hinges.68
6.16. Total stress due to load combination with truck in position 3 with three hinges.70
6.17. Total stress due to load combination with truck in position 4 with three hinges.73
6.18. Dead load values in Risa FE..............................................76
viii
7.1. Results from MEXE method and Ring.
83


LIST OF FIGURES
FIGURE
2.1. Distribution forces in the arch..................................................5
2.2. Corbelled arch...................................................................5
2.3. Fixed-fixed arch.................................................................7
2.4. Two-hinged arch..................................................................7
2.5. Three-hinged arch................................................................7
2.6. Important parts of an arch bridge................................................8
2.7. Arch and parapet in bridge.......................................................9
3.1 A photograph of the bridge when it was registered as a historic place..........12
3.2. Satellite view of the Ely Stone Bridge.........................................13
3.3. Side view of the Ely Stone Bridge at east side..................................13
3.4. Side view of the Ely Stone Bridge at west side..................................14
4.1. Three hinged arch...............................................................16
4.2. Arch bridge dimensions for MEXE method (Robinson BEng, 2000)................... 18
4.3. Shows explanations a RING2.0 analysis (limitState, 2009)....................... 19
5.1. Illustration of arch barrel strip...............................................21
5.2. Weight of fill along the strip of arch barrel...................................22
5.3. Live load “truck 1” location....................................................22
5.4. Live load “truck 2” location....................................................23
5.5. Live load “truck 3” location....................................................23
5.6. Live load “truck 4” location....................................................24
5.7. Load combination with truck at position 1......................................25
IX


5.8. Load combination with truck at position 2
26
5.9. Load combination with truck at position 3.......................................27
5.10. Load combination with truck at position 4......................................28
6.1. Stress of strip of arch barrel due to dead load (self-wight, fill)..............29
6.2. Stress of strip of arch barrel due to live load (truck 1)........................30
6.3. Stress of strip of arch barrel due to live load (truck 2)........................30
6.4. Stress of strip of arch barrel due to live load (truck 3)........................31
6.5. Stress of strip of arch barrel due to live load (truck 4)........................31
6.6. Stress of strip of arch barrel due to load combination with truck at position 1..32
6.7. Stress of strip of arch barrel due to load combination with truck at position 2..32
6.8. Stress of strip of arch barrel due to load combination with truck at position 3..33
6.9. Stress of strip of arch barrel due to load combination with truck at position 4..33
6.10. Stress of strip of arch barrel due to dead and live loads......................34
6.11. Arch dimensions (Robinson, 2000)............................................... 34
6.12. The Ely Stone Bridge dimensions................................................35
6.13. Nomogram for determining the provisional axle loading of masonry arch bridges before
factoring (The highways agency, 2001)................................................ 37
6.14. Span/Rise factor (The highways agency, 2001)................................... 38
6.15. Profile factor (The highways agency, 2001)..................................... 39
6.16. The First position of the truck load in Ring...................................44
6.17. The last position of the truck load in Ring.....................................44
6.18. Model of the Ely Stone Bridge in Ring...........................................45
6.19. Model of the Ely Stone Bridge in Ring (3D view).................................45
6.20. Moment diagram for one span assuming continuity of all stone unites.............46
x


6.21. Normal axial force diagram assuming continuity of all stone unites...............46
6.22. Joint coordinates for Risa 2D Frame Analsis......................................47
6.23. Self-weight load on the arch.....................................................49
6.24. Fill load on the arch............................................................50
6.25. Pressure load on the right side of the arch......................................50
6.26. Pressure load on the left side of the arch.......................................51
6.27. Axial force (kips) from dead load................................................51
6.28. Moment (kips-ft) from dead load..................................................52
6.29. The location of truck 1..........................................................54
6.30. Axial force (kips) due to load combination of dead load and live load for truck in
position 1..............................................................................54
6.31. Moment (Kips-ft) due to load combination of daed load and live load for truck in
position 1..............................................................................55
6.32. The location of truck 2..........................................................57
6.33. Axial force (kips) due to to load combination of daed load and live load for truck in
position 2..............................................................................57
6.34. Moment (kips-ft) due to to load combination of daed load and live load for truck in
position 2..............................................................................58
6.35. The location of truck 3..........................................................60
6.36. Axial force (kips) due to to load combination of daed load and live load for truck in
position 3..............................................................................60
6.37. Moment (kips-ft) due to to load combination of daed load and live load for truck in
position 3..............................................................................61
6.38. The location of truck 4..........................................................63
xi


6.39. Axial force (kips) due to to load combination of daed load and live load for truck in
position 4............................................................................63
6.40. Moment (kips-ft) due to to load combination of daed load and live load for truck in
position 4............................................................................64
6.41. Model of the arch with hinges in Risa 2D Frame.................................65
6.42. Axial force (kips) due to load combination of dead load and live load for truck in
position 1 with three hinges.........................................................67
6.43. Moment (kips-ft) due to load combination of dead load and live load for truck in
position 1 with three hinges.........................................................67
6.44. Axial force (kips) due to load combination of dead load and live load for truck in
position 2 with three hinges..........................................................69
6.45. Moment (kips-ft) due to load combination of dead load and live load for truck in
position 2 with three hinges..........................................................70
6.46. Axial force (kips) due to load combination of dead load and live load for truck in
position 3 with three hinges.........................................................72
6.47. Moment (kips-ft) due to load combination of dead load and live load for truck in
position 3 with three hinges.........................................................72
6.48. Axial force (kips) due to load combination of dead load and live load for truck in
position 4 with three hinges.........................................................74
6.49. Moment(kips-ft) due to load combination of dead load and live load for truck in
position 4 with three hinges..........................................................75
6.50. Finit element models............................................................75
6.51. Self-weight load Fill load in Risa FE..........................................77
6.52. Pressure load on the right side of the arch in Risa FE.........................77
6.53. Pressure load on the left side of the arch in Risa FE..........................78
xii


6.54. Truck 1 position in Risa FE............................................................78
6.55. Truck 2 position in Risa FE............................................................79
6.56. Truck 3 position in Risa FE............................................................79
6.57. Truck 4 position in Risa FE............................................................80
6.58. The stress by dead load in Risa FE......................................................80
6.59. The stress by dead load and live load “truck 1” in Risa FE............................81
6.60. The stress by dead load and live load “truck 2” in Risa FE............................81
6.61. The stress by dead load and live load “truck 3” in Risa FE............................82
6.62. The stress by dead load and live load “truck 4” in Risa FE............................82
7.1. Model of the Ely Stone Bridge in Risa 2D Frame............................................84
7.2. Model of the Ely Stone Bridge in Ring.....................................................85
7.3. Axial force (kips) diagram based on Ring’s 220 kips axle load, in Risa 2D................85
7.4. Moment diagram by the critical design load in Risa 2D....................................86
7.5. Failure mechanism for the The Ely Stone Bridge from RING2.0.............................86
7.6. Axial force diagram by the critical design load in Ring...................................87
7.7. Moment diagram by the critical design load in Ring.......................................87
7.8. Model of the Ely Stone Bridge under critical load in Risa 2D Frame with three hinges. 88
7.9. Axial force (kips) diagram by the critical design load in Risa 2D with three hinges......88
7.10. Moment (kips-ft) diagram by the critical design load in Risa 2D with three hinges.......89
xiii


CHAPTER I
1 INTRODUCTION
1.1 Introduction
Stone masonry arch bridges are one of the oldest forms of structures. Many have performed for centuries. Many historic stone masonry arches now carry live loads much greater than their original intended purpose (Lily Beyer, 2012).
The U.S. has far fewer stone arch bridges than European countries. Many of these stone arch bridges were built prior to 1910; one of these is the Ely Stone Bridge in Jones County, Iowa. Its builders could not have predicted the modern-day changes of width, truck load, guard rails, drainage, erosion, and the long-term effects of flooding and freeze/thaw cycles.
1.2 Research Objective
The main goal of this thesis is to study and assess the Ely Stone Arch Bridge in light of current traffic loads, and to apply multiple analytical methods to the Ely Stone Bridge. The bridge will be modeled analytically using the following methodologies:
• The MEXE Method
• Ring 3.2
• Risa 2D Frame Analysis
• Risa 2D Finite Element Method (FEM)
1.3 Outline of This Thesis
The contents of each chapter are briefly described below:
Chapter I: This chapter will show an overview of the masonry arch bridge, statement of the research objective, and outline of this thesis.
1


Chapter II: This chapter will present a brief a literature review.
Chapter III: This chapter will have description of the Ely Stone Bridge.
Chapter IV: This chapter will discuss different types of assessment software such as MEXE method, Ring method, Risa 2D Frame method, and Risa 2D Finite Element method.
Chapter V: This chapter will present the loads that are applied on the arch and parameters of the bridge.
Chapter VI: This chapter will illustrate the analysis of the bridge resulting from different methods.
Chapter VII: This chapter will provide a comparison of different calculation methods, conclusion, and recommendations for future work.
2


CHAPTER II
2 LITERATURE REVIEW
2.1 Introduction
There are approximately 1700 masonry stone arch bridges in the United States according to 2013 National Bridge Inventory database (NBI, 2013). Approximately, 50% of these bridges were constructed before 1910 and approximately 50% of them are still in service more than 100 years later. On the other hand, the same study shows that only 4% of the steel bridges, and 1% of the concrete bridges built before 1910 remain in service. Since most early settlers in the United States were of European heritage, they had the skills required to build arch bridges (Citto & Woodham, 2016).
2.2 History of Stone Masonry Arch Bridge
Bridges are an important part of the transportation infrastructure. Many centuries ago, the masonry stone arch bridge was a common form of construction. The original form of stone masonry bridges goes back to ancient times, most notably the Romans (Boyd, 1978).
The Greeks were the first to use the arch in structures. There is an example of a Greek arch bridge which is a very small bridge in Rhodes. On the other hand, the Romans built great arch bridges and used them as aqueducts to transfer water to their cities (Boyd, 1978).
In the Middle Ages, after the fall of the Roman Empire, building bridges became more important than ever before. The reason for this development is related to armies needing bridges to move troops and supplies. However, with this increased construction of bridges, the engineering knowledge, and economic resources required to build these bridges became more commonplace (Boyd, 1978).
3


At the present time, there are many bridges, built in the nineteenth century and earlier, that are still functioning and still serving critical roles in roadway and railway networks around the world. However, these bridges are likely carrying traffic loads much higher than the load for which they were originally intended (Boyd, 1978).
Even though the structural behavior of masonry arch bridge appears simple and easy at first, their modelling and mechanical behavior is complicated because of nonlinear effects. In addition, even if no new masonry arch bridges are built, the restoration of existing bridges is complicated by the use of both old and new materials and techniques (Boyd, 1978).
2.3 Arches
The arch design is one of the oldest forms of bridges and has great strength. Before 20th century, the arches were used to span the distance between two supports such as walls or piers and they were constructed by using stone and bricks. In the 20th century, arches are commonly used in bridge construction and are constructed using steel and reinforced concrete.
Arches are resistant to loads that include live load and self-weight of the bridge to the abutments at each end, where those ends resist the horizontal thrust. So, the abutments of the arch should be large to resist these forces.
However, masonry arches need to be designed for stability because they are subjected to separation of joints in regions of high flexural tension.
4


I I I
Figure 2.1. Distribution forces in the arch.
A curved beam faces horizontal and vertical reactions. So, the support at the two ends must not be a roller. These types of arches do not develop horizontally at the base like a corbelled arch. A corbelled arch was common in ancient civilizations and in the Americas. This kind of arch can develop bending stress in its members (Drysdale and Hamid, 2008).
Figure 2.2. Corbelled arch.
Corbelling masonry dates back to 2900 B.C. An example of a corbel arch can be found in the Mycenaean fortress at Tiryns, Greece. Corbelling uses subsequent courses of stone or brick on each side of the opening, where each course protrudes beyond the previous
5


course. These courses of protruding materials were supported until the entire arch was completed (Drysdale and Hamid, 2008).
The arch shape is important for stone masonry bridges because this form can easily develop compressive stress. Compressive stress is a positive feature because the stones in the arch are effectively prestressed in compression. For this reason, it was a preferred shape for a long time (Beyer, 2012).
The shape of the arch was made of stones which were cut into a trapezoidal form. The keystone is important component in creating the arch shape. In addition, the arch can be made from bricks bonded with mortar. This type of arch is common in many countries especially in the United Kingdom.
2.3.1 Arches classified.
There are three types of arches which are usually used:
> Fixed-Fixed Arch
> Two-Hinged Arch
> Three-Hinged Arch
Fixed-fixed arches and two-hinged arches are statically indeterminate structures. However, the reactions and internal forces can be calculated by flexibility matrix method.
The three-hinged arch is a statically determinate structure which means that the reactions and all internal forces can be calculated by static equations of equilibrium (Three Hinged Arch).
6


7


2.3.2 Behavior of the arch under load.
When dead load is applied, the arch will be very stable, especially if the centerline of the arch is very close to the line of thrust. In large arches, the live load is smaller than the dead load. So, the effects of live load will be limited. On the other hand, live load will be more predominate than dead load on small arch structures.
When an unbalanced live load is applied on the arch, it will cause a bending moment in the rib, or barrel of the arch. In this situation, the arch must be designed to resist the additional load so the arch will remain stable under the unbalanced load (Lily Beyer, 2012).
2.4 Description of the Stone Masonry Arch Bridge
Masonry arch bridges typically have different components that include arch, abutment, backfill, and the wing walls. It also has one or more rings of stone as seen in Figure 2.6.
Figure 2.6. Important parts of an arch bridge.
As shown in Figure 2.7, a masonry arch bridge has two basic component elements which are the arch and parapet.
8


Figure 2.7. Arch and parapet in bridge.
Stone is the main component in many masonry arch bridges but it is very weak in tension. Also, joints between stone units tend to open when in tension. So, the material used in masonry arch bridges, has very low tensile strength and has great ability to resist compression. The strength in masonry arches is dependent on the strength and stability of arch barrel.
2.4.1 Materials.
Stone bridges are often made from sandstone or limestone. Sandstone is sedimentary debris rock and it has resistance to compression between 3.5 to 14.5ksi. Limestone is carbonated rock and it has resistance to compression between 3 to 11.5ksi (Martinez et al, 2001).
2.5 Structural Analysis of Old Masonry Arch Bridge
The structural analysis of old masonry bridges is not an easy or simple matter. These bridges were intended for live loads common in their construction time. However, they are now carrying loads beyond those envisioned by their designer.
9


There are many reasons that make modeling this type of bridge difficult, e.g. the behavior of masonry arches is not commonly found in some software. Also, the information and history of this bridge may not be complete.
10


CHAPTER III
3 THE ELY STONE BRIDGE
3.1 Introduction
Stone masonry arch bridges are one of the oldest forms of structures. This kind of bridge has many issues and questions related to the modeling and analysis. In addition, most of these bridges increasingly deteriorate. For instance, the Ely Stone Bridge in Jones County, Iowa, is a 100 year old stone masonry bridge located in the United States. It has successfully served for many years. This bridge has several condition issues; however, it is still in service, and it remains as a part of the history of the county. It is only natural for a community to want to preserve the beautiful and functional Ely Stone Bridge given its history and heritage in the town of Monticello, Iowa.
3.2 Background and Description
In 1893, Reuben Ely Sr. and his son Reuben Ely Jr. constructed the Ely Stone Bridge, which has three elliptical stone arches, near Monticello in Jones County, Iowa. They used native stone which was brought in from different places in Iowa. The limestone used for the voussoir was brought in from Anamosa, Iowa, and the stone that is located on the side spandrels and piers was brought in from streambeds. The stone from Anamosa, Iowa, which was used to build the arches of the bridge, was located about twelve miles from the construction site. The Ely Stone Bridge measures approximately 68 feet long, 15.5 feet wide and crosses over the Wet Creek. Also, it has three spans of masonry elliptical arches. Each span measures approximately 20 feet in length and each arch span has double layers of limestone units called voussoirs. There are vertical piers which are approximately 5 feet high.
11


Like a modem bridge, the top of the Ely Stone Bridge has pavement which is a reinforced
concrete slab. It is approximately 8 inches thick and around 16 feet wide.
In addition, the Ely Stone Bridge has been listed in the National Register of Historic Places in 1979 (NRHP 1979, Jackson 1988).
Figure 3.1 A photograph of the bridge when it was registered as a historic place.
12


Figure 3.2. Satellite view of the Ely Stone Bridge.
Photographs of the Ely Stone Bridge are shown in Figures 3.3 and 3.4 below:
Figure 3.3. Side view of the Ely Stone Bridge at east side.
13


Figure 3.4. Side view of the Ely Stone Bridge at west side. (For more detailed information about the Ely Stone Bridge, see Appendix A)
14


CHAPTER IV
4 ANALYSIS METHODS: GENERAL
4.1 Introduction
There are numerous ways to model and analyze a stone arch bridge. These methods have been developed since antiquity, through the inception of modem structure analysis in the 19th century, and continue to this day. In the past, the methods for structural assessment were based on approximate calculation methods. They only provide an approximate estimate of the capacity load. After years, tremendous progress provided more advanced numerical modelling strategies, and the analysis, description, and geometry of the bridge was more accurate. Advances in technology developed new software which can create models using finite element modeling methods. There are two and three-dimensional finite element models. In this chapter, various methods are presented and compared their results. However, these methods are not a complete inventory.
4.2 Methods Used for Modeling
There are two different levels of analysis used in this paper to examine the condition of the Ely Stone Bridge. The first level is defined as the approximate calculations method and it still widely used. This level of analysis contains methods such as:
• The graphical analysis method
• The three-hinged arch method
• The MEXE method.
At the second level, there is a simple 2D modelling such as:
• The Risa 2D Frame Analysis method
• The Risa 2D Finite Element method
15


• The Ring method
4.2.1 Three-Hinged Arch method
From the name of this method, it is clear to understand that Three-Hinged Arch method based on three hinges. There are two-hinges at the supports, which are located on the end of the arch, and the third hinge will be the support at the crown of the arch.
This method is based on the calculation of the bending moment in a three-pinned arch. It is a simple and easy method because the reactions on the arch can be determined by using the forces on each direction equal to zero. A three-pinned based on simple statics method. The stress in the member can be calculated by determined the bending moment in this member (Three Hinged Arch).
In order to calculate any internal moment at any point on an arch, the reaction on the abutments must first be calculated. In this type of arch, there are four reactions and all of them are unknown. However, there are three static equations equilibrium and one moment equation. The moment equation, determine the moment around the third hinge for all forces on the arch, on either side of it, to calculate the reactions (Three Hinged Arch).
Figure 4.1. Three hinged arch. 16


4.2.2 The Modified MEXE method
The Modified MEXE (henceforth referred to as “MEXE”) method is commonly used, particularly in the ETC. This method was developed by the Military Engineering Experimental Establishment, MEXE. The MEXE method is as a considered approximate calculation method. However, it is fast and easy in comparison with other methods (Lasell and Bjurstrom, 2009).
There are some limitations in order to be able to use this method:
• The arch should not be appreciably deformed or have a flat shape.
• The span of bridge must not be longer than 18m (59ft).
• The skew of the bridge must be smaller than 15 degrees.
• The backfill above the extrados has to be less than lm (3.3ft).
• The bridge does not have a multi- span (except if the ratio of the arches work separately and the ratio of the height to width piers of multi span is smaller than 2).
The target of the MEXE method is to calculate the provisional axle load (PAL) capacity of masonry arches. So, PAL based on the span, the average depth between the arch at the crown and the surface of road, and the thickness of the arch barrel (Robinson BEng, 2000). The following Definition of arch bridge dimensions as shown in Figure 4.2.
L...............The span (m)
rc.............. The rise of the arch at the crown (m)
rq.............. The rise of the arch at the quarter points (m)
d...............The thickness of the arch barrel to the keystone (m)
h.............. The depth of the fill (m)
17


parapet level
l
Figure 4.2. Arch bridge dimensions for MEXE method (Robinson BEng, 2000).
4.2.3 RING 3.2
Ring 3.2 is software which can assess single and multiple span masonry arch bridges. It can calculate the maximum safe load that can be applied on the arch bridge. Also, it is able to determine the critical failure load that impacts the arch bridge. Ring considers the bond between the blocks with the help of friction coefficients. The analysis method used in Ring can work with bridges that face little damage, bridges that have lost their mortar, and masonry material. The program shows the thrust line with failure model at the same time (LimitState, 2009).
18


Figure 4.3. Shows explanations a RING2.0 analysis (limitState, 2009).
4.2.4 RISA
Risa is a suite of programs that are used widely in structural analysis and design in the
United States. Risa software can be used for analysis of tunnels, roller coasters, soccer stadiums, buildings, etc. In this study, Risa 2D Frame analysis and Risa 2D Finite Element
analysis are used (RISA, 2013).
4.2.5 Three-dimensional FEM
A masonry arch bridge can be modeled in three dimensions. This analysis can model the entire structure which included abutments, spandrel walls. We might expect this kind of analysis to provide more realistic results and give better load carrying capacity. However, three- dimensional finite element (FEM) analysis is difficult to use for masonry unites when joint separations occur.
19


CHAPTER V
5 LOADS
5.1 Introduction
Loads largely affect the analysis of a structure. Applying different dead loads over a structure and different locations of live loads can largely effect the design. In this study, dead load, live load, and earth pressure load have been considered. Neither snow load nor temperature load were considered.
5.2 Design Codes and Regulations
In the United States, AASHTO LRFD 2012 Bridge Design Specifications is typically used for bridge design.
5.3 Loads
Dead load, live load, and earth pressure load have been applied to analyze of the Ely Stone Bridge. For this study, live load was applied at four positions on the bridge. Based on that, five load combinations were used:
> Dead load only
> Dead plus truck location 1
> Dead plus truck location 2
> Dead plus truck location 3
> Dead plus truck location 4
All loads will be applied on the strip of arch barrel as shown in Figure 5.1 below:
20


Figure 5.1. Illustration of arch barrel strip.
5.3.1 Dead load.
Dead load includes the weight of structure, fill load, and lateral pressure load.
Lateral pressure load from the fill was computed based on the weight of fill, it’s variable depth, and a coefficient of active earth pressure of 0.3. In fact, the arch weight leads to the creation of stress into its members. The self- weight of the arch has been included in all analysis. The stresses that are created from dead load is mostly compression stress with some bending stress in the arch.
To calculate fill load along the arch, the depth for each point along the x- axis of the arch multiplied by fill's self-weight (y): 0.120 kips per cubic foot (kef). Because the depth of the arch changes over the length of the bridge, this causes change in the value of fill load as seen in Figure 5.2.
21


Figure 5.2. Weight of fill along the strip of arch barrel.
5.3.2 Live load “truck load”
Live load is a variable load, and it induces the maximum stress when it is located over only part of the span. In this paper, live load was applied as a distributed load by the area on the strip of the arch barrel. However, it will be covered specific area on the strip depended the depth of the fill. The truck load was applied in four positions as shown in Figures below:
TRUCK
12.00'
Figure 5.3. Live load “truck 1” location.
22


Figure 5.4. Live load “truck 2” location.
Figure 5.5. Live load “truck 3” location.
23


TRUCK 4r-
22.02'
Figure 5.6. Live load “truck 4” location.
5.4 Load Combinations
Combining all loads, which include self-weight of the arch, fill load, pressure load, and live load, will create stresses as shown in Figures below:
5.4.1 Combined dead load and live loads
The load combination which includes self-weight, fill load, live load (Truck load), and pressure load. The distribution of all loads on all the arch shown in Figures below:
24


This load combination will have self-weight, fill load, live load (truck position 1), and
pressure load as seen in Figure 5.7.
5
25
Figure 5.7. Load combination with truck at position 1.


This load combination will have self-weight, fill load, live load (truck position 2), and
pressure load as seen in Figure 5.8.
26
Figure 5.8. Load combination with truck at position 2.


This load combination will have self-weight, fill load, live load (truck position 3), and
pressure load.
27
Figure 5.9. Load combination with truck at position 3.


This load combination will have self-weight, fill load, live load (truck position 4), and
pressure load.
28
Figure 5.10. Load combination with truck at position 4.


CHAPTER VI
6 ANALYSIS AND COMPARSION OF THE RESULTS
The goal of this paper is to compare different methods for analyzing masonry stone arch bridges. The comparisons have been made with different methods to calculate the ultimate load capacity using the same arch, under the same loads.
6.1 Linearly Elastic Analysis
From the loads described in chapter 5, and assuming linear elasticity applies (i.e. no hinges). Then stresses in the barrel of the arch can be determined.
To calculate the stress in each section of the arch due to dead load we need to know shape section, axial force, and the moments that are created by dead load for each of the node on the arch. The summation of the axial force is divided by the area of the stone and the moment divided by the section modulus. From Figure 6.1, the maximum stress occurs close to the ends, where the arch cross-section area is under the maximum dead load.
Figure 6.1. Stress of strip of arch barrel due to dead load (self-wight, fill).
In addition, the stress in each segment on the arch due to live load at each position, the live load is distributed as a concentrated load on the top of the arch to a distributed load
29


on the strip of the arch barrel. The moment and axial force along the strip of the arch are calculated to determine the stresses on all sections.
The following Figures illustrate the stresses due to truck load on the strip of the arch at four positions.
Figure 6.2. Stress of strip of arch barrel due to live load (truck 1).
Figure 6.3. Stress of strip of arch barrel due to live load (truck 2).
30


0.050
-0.030 J----------1---------1----------1-----
length of the arch (ft)
Figure 6.4. Stress of strip of arch barrel due to live load (truck 3).
Figure 6.5. Stress of strip of arch barrel due to live load (truck 4).
31


Stress
The stresses of all loads combination shown in Figures below:
Figure 6.6. Stress of strip of arch barrel due to load combination with truck at position 1.
Figure 6.7. Stress of strip of arch barrel due to load combination with truck at position 2.
32


Figure 6.8. Stress of strip of arch barrel due to load combination with truck at position 3.
Figure 6.9. Stress of strip of arch barrel due to load combination with truck at position 4.
33


0.100
0.080
0.060
0.040
Xfl m CD 0.020
IS1 0.000
-0.020
-0.040
length of the arch (ft)
â– Load combination with truck 1 â– Load combination with truck 2 â– Load combination with truck 3 Load combination with truck 4
Figure 6.10. Stress of strip of arch barrel due to dead and live loads.
6.2 Analysis Using MEXE
The modified MEXE method is used to determine the carrying capacity of arches. The first step in this method is to calculate PAL, then it must be factored depending on material factor, joint factor, profile factor, and span factor. Dimensions used in the MEXE method analysis are shown in Figure 6.11.
Figure 6.11. Arch dimensions (Robinson, 2000).
34


L.................The length of the span (m)
rc................Ratios to intrados of semi-circular arch (m)
rq................Rise at L/4 (m)
d.................The arch depth (m)
h.................Fill above arch (m)
This result of the MEXE method for the Ely Stone Bridge based on the dimensions that are shown in the Figure 6.12 below:
Where
L= 19.82ft= 6.0 m
rc =5.6ft = 1.7 m rq = 3ft = 0.9 m h = 2.15ft = 0.7 m d= 2ft= 0.6 m
35


6.2.1 Calculate the provisional axle load.
The provisional axle loading can be obtained by following equation:
PAL = 740 <70 (tonnes)
Note : 1 (metric) tonne = 1.023 (U.S) ton.
The provisional axle load can be determined by the nomogram by simply drawing three straight lines (A, B, and C) as shown in Figure 6.3 below. Mark the (L), which is arch span, on column A, and mark (d + h), which is the total crown thickness (barrel and fill), on column B then draw line to column C. The number that faces the line will be the PAL in tons. Even by using the equation or the nomogram, the result has to be the same (The highways agency, 2001).
By using the equation:
PAL = 740 (°'6+°137)2= 114.3 > 70
(6.0)1-3 -
Use PAL= 70.0 tonnes
This value of the provisional axle load must be modified by multiple factors.
By using the following nomogram
L=6.0 m
d+h=1.3m
Line through these points to column C PAL= 70.0 tonnes
36


A
ARCH SPAN
(L)
m
18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -
6 -
5 -
4.5 -4 -
3.5 -
3 -
2.5 -
B
TOTAL CROWN THICKNESS
(h + d)
m
1.8-
1.6-
1.4-
1.2-
1.0-
0.9-
0.8-
0.7-
0.6-0.55-0.5-0.45-0.4 -0.35-0.3-
0.25-
c
PROVISIONAL AXLE LOADING
(P.A.L.) TONNE 70 —i
60-
50-
42-
36-
30-
27-
24-
21 -
18 -
15-
12-
9-
2 -
1.5 -1
6 -
Figure 6.13. Nomogram for determining the provisional axle loading of masonry arch bridges before factoring (The highways agency, 2001).
37


6.2.1.1 Calculate modification factors.
1. Span/Rise Factor (Fsr): this represents the arch shape. When a span/rise ratio is 4 or less, then it will be assumed to give optimum strength and the factor assumed to be “1”. However, if the ratio between span and rise is greater than 4, Figure 6.14 will be used to give the appropriate span/rise factor (Fsr). Where the equation of span/rise factor calculated by following expression (The highways agency, 2001).
Fsr = —
rc
Span/Rise Ratio Lyr c
Figure 6.14. Span/Rise factor (The highways agency, 2001).
Fsr = - = — =3.5 <4.0
rc 1.7 “
Then take Fsr = 1.0
2. Profile Factor (Fp): this factor is based on the fact that segmental and parabolic arches are stronger than elliptical arches of similar span/rise ratio and thickness of barrel. The profile factor (Fp) can be calculated even by Fig 6.15 or by the following equation (The highways agency, 2001).
38


Fp = 2.3 0,6
rc
Fp = 1.5 ThakeFp = 1
Also, from Figure below, — = 0.5 , Fp= 1.0
Figure 6.15. Profile factor (The highways agency, 2001).
3. Material Factor (Fm): This factor focuses on the depth of the fill and the height of the barrel. The material factor calculated by equation below (The highways agency, 2001).
Fm _ (Fpxd)+(F/xft) d+h
Where the values for Fp andF/can be determine from Tables 6.1 and 6.2 respectively.
39


Table 6.1. Barrel factor (The highways agency, 2001).
Arch Barrel Barrel Factor (Fb)
Granite whether random or coursed and all built-in-course masonry except limestone, all with large shapes voussoirs 1.5
Ashlar quality siliceous sandstone 1.4
Concrete # or engineering bricks and similar sized masonry (not limestone). 1.2
Limestone, whether random or coursed, ashlar quality calcareous sandstone, good random masonry and building bricks, all in good condition. 1.0
Masonry of any kind in poor condition (many voussoirs flaking or badly spalling, shearing etc.). Some discretion is permitted if the dilapidation is only moderate. 0.7
From Table 6.1, appropriate values of the barrel factor Fb = 1.0
Table 6.2. Fill factor (The highways agency, 2001).
Filling The Fill Factor (Ff)
Concrete # 1.0
Grouted materials (other than those with a clay content) 0.9
Well compacted materials* 0.7
Weak materials evidenced by tracking of the carriageway surface 0.5
And from Table 6.2, appropriate values of the fill factor Ff= 0.7 Apply these values in the last equation and determine Fm Fm =1.0
4. Joint Factor (Fi): this factor is based on the size and situation of joints and general size. It is calculated from the following formula (The highways agency, 2001).
40


Fj = FwxFdxFmo Where:
Fw = is the width factor Fd= is the depth factor Fmo =is the mortar factor.
All these factors can be acquired from Tables 6.3, 6.4, and 6.5 respectively.
Table 6.3. Width factor (The highways agency, 2001).
Width of Joint Width Factor (Fw)
Joints with widths up to 6mm 1.0
Joints with widths between 6mm and 12.5mm 0.9
Joints with widths over 12.5mm 0.8
From Table 6.3, appropriate values of the width factor Fw = 0.8
Table 6.4. Depth factor (The highways agency, 2001).
Construction of Joint Depth Factor (Fa)
Unpointed joints, pointing in poor condition and 0.9#
joints with up to 12.5mm from the edge insufficiently filled
Joints with from 12.5mm to one tenth of the thickness of the barrel insufficiently filled 0.8#
Joints insufficiently filled for more than one tenth At the +
the thickness of the barrel engineer’s discretion
From Table 6.4, appropriate values of the depth factor Fd= 1.0
41


Table 6.5. Mortar factor (The highways agency, 2001).
Condition of Joint Mortar Factor ( Fmo )
Mortar in good condition 1.0
Loose or friable mortar 0.9
From Table 6.5, appropriate values of the mortar factor Fmo= 1.0 Fj = FwxFdxFmo =0.8x1.Oxl.O =0.8
5. Condition Factor (Fern): This factor is based on an assessment of the importance of the various cracks and deformations. The range of this factor is between 0 and 1.0 based on the stability and load carrying capacity of the arch barrel. Where Fern equal 1.0 when the arch has good condition with no defects. However, Fern equal 0 when the bridge has poor condition (The highways agency, 2001).
Fcm =1.0
6.2.2 Modified axle load.
To determine the modified axle load that shows the allowable loading on the arch, we must apply all previous factors and multiply by the provisional axle loading as given in following expression (The highways agency, 2001).
Wm = FsrxFpxFmxFjxFcmxPAL
= 1.0x1.0x1.0x0.8x1.0x70
= 54 tonnes =120 kips
For more information about this method, see Appendix B.
42


6.3 Analysis Using Ring.
RING 3.2 software analysis was performed for the Ely Stone Bridge. The Ely Stone Bridge was modelled with 18 ft (5791 mm) width, 19.82 ft (6000 mm) a span arch bridge, 2.15 ft (700 mm) a fill height over arch crown, and 5.6 ft (1707 mm) the rise of the arch at the crown. In addition, the bridge was analyzed with the parameters that are presented in Table 6.6 below:
Table 6.6. Parameters used in Ring for the Ely Stone Bridge.
Material Properties Masonry Backfill
Unit Weight (KN/m3) 25 19
Friction coefficient 0.6 0.6
Angle of internal friction " 30
A single axle-loads was applied spread apart to make sure our results were positioned over the span with value of 1 KN and again with a value of 112.82 KN. They gave the same critical load at the same location. However, all load information is presented in Table 6.7. The critical design load happened with a single-axle load as shown below:
Table 6.7. Load factor used in Ring for the Ely Stone Bridge.
Load kind Adequacy factor Critical design load Position of the load from support
Single Axle (lkN) 977 220 kip 3900 mm
EU Single Axle 8.66 220 kip 3900 mm
(112.82 KN)
43


Figure 6.16. The First position of the truck load in Ring.
Figure 6.17. The last position of the truck load in Ring.
44


Figure 6.18. Model of the Ely Stone Bridge in Ring.
Figure 6.19. Model of the Ely Stone Bridge in Ring (3D view).
45


Figure 6.20. Moment diagram for one span assuming continuity of all stone unites.
Figure 6.21. Normal axial force diagram assuming continuity of all stone unites.
To see more details of this method, Appendix C provides all analysis results for one span and three spans.
46


6.4 Analysis Using Risa 2D Frame.
Risa 2D Frame analysis is based on plane frame analysis. Using the frame analysis means using the simplest form of modelling. In this analysis, the frame usually used fixed supports on the abutments.
The Ely Stone Bridge was modelled with different joint coordinates on a strip of arch barrel. Where the strip was divided into many elements based on different dimensions in X and Y as shown in Figure below
R
V 2 *
£
•A
9
2,789
4.704
6.97
9.478
12.11
14.742
17.25
19.516
21.431
22.906
24.22
Figure 6.22. Joint coordinates for Risa 2D Frame Analsis.
47


Also, the bridge performed with the parameters illustrated in the Table below: Table 6.8. Parameters used in Risa 2D Frame for the Ely Stone Bridge.
Material Properties Arch (Limestone)
E- Elastic Modulus (ksi) 4400
G- Shear Modulus (ksi) 1760
u - Poisson's Ratio 0.25
y - Stone’s Self Wight (kef) 0.160
Y - Fill's Self Wight ( kef) 0.120
6.4.1 Loads
The loads applied on the arch are dead load, which include self-weight load, fill load, and pressure load, and live load, which include truck load on four different positions as shown in the Table and Figures below:
Table 6.9. Dead load values in Risa 2D Frame.
Node# Self-Wight Load (k/ft) Fill Load (k/ft) Pressure Load (Right) (k/ft) Pressure Load (Left) (k/ft)
N1 0.32 1.17 0.351
N2 0.32 0.92004 0.276012
N3 0.32 0.738 0.2214
N4 0.32 0.63 0.189
N5 0.32 0.5508 0.16524
N6 0.32 0.498 0.1494
N7 0.32 0.45 0.135
N8 0.32 0.4248 0.12744
N9 0.32 0.405 0.1215
N10 0.32 0.39 0.117
Nil 0.32 0.38004 0.114012
N12 0.32 0.375 0.1125
N13 0.32 0.38004 0.114012
48


N14 0.32 0.39 0.117
N15 0.32 0.405 0.1215
N16 0.32 0.4248 0.12744
N17 0.32 0.45 0.135
N18 0.32 0.498 0.1494
N19 0.32 0.5508 0.16524
N20 0.32 0.63 0.189
N21 0.32 0.738 0.2214
N22 0.32 0.92004 0.276012
N23 0.32 1.17 0.351
Figure 6.23. Self-weight load on the arch.
49


Figure 6.24. Fill load on the arch.
Figure 6.25. Pressure load on the right side of the arch.
50


Figure 6.26. Pressure load on the left side of the arch.
The Figures below show the results which are axial force and moment from dead
load:
Figure 6.27. Axial force (kips) from dead load.
51


Figure 6.28. Moment (kips-ft) from dead load.
The RISA 2D frame analysis uses four different load combinations based on four positions of the live load. The description and result for each load combination is shown on the Tables below:
6.4.1.1 Load combination with truck in position 1
This load combination is self-weight load, fill load, pressure load, and live load for
truck 1.
Table 6.10. Total stress due to load combination with truck in position 1.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress (ksi) Total Stress (psi)
Ml I End 13.0 -3.17 0.05 0.03 -0.03 0.01 12
j End 12.3 2.09 0.04 -0.02 0.02 0.06 65
M2 I End 11.9 2.09 0.04 -0.02 0.02 0.06 63
j End 11.4 5.08 0.04 -0.05 0.05 0.09 93
M3 I End 11.2 5.08 0.04 -0.05 0.05 0.09 92
j End 10.9 4.71 0.04 -0.05 0.05 0.09 87
52


M4 I End 10.6 4.71 0.04 -0.05 0.05 0.09 86
j End 10.4 3.57 0.04 -0.04 0.04 0.07 73
M5 I End 10.0 3.57 0.03 -0.04 0.04 0.07 72
j End 9.9 1.62 0.03 -0.02 0.02 0.05 51
M6 I End 9.8 1.62 0.03 -0.02 0.02 0.05 51
j End 9.6 0.23 0.03 0.00 0.00 0.04 36
M7 I End 9.3 0.23 0.03 0.00 0.00 0.03 35
j End 9.3 -1.96 0.03 0.02 -0.02 0.01 12
M8 I End 9.2 -1.96 0.03 0.02 -0.02 0.01 12
j End 9.2 -3.76 0.03 0.04 -0.04 -0.01 -7
M9 I End 9.2 -3.76 0.03 0.04 -0.04 -0.01 -7
j End 9.1 -5.18 0.03 0.05 -0.05 -0.02 -22
M10 I End 9.2 -5.18 0.03 0.05 -0.05 -0.02 -22
j End 9.1 -6.06 0.03 0.06 -0.06 -0.03 -31
Mil I End 9.2 -6.06 0.03 0.06 -0.06 -0.03 -31
j End 9.1 -6.21 0.03 0.06 -0.06 -0.03 -33
M12 I End 9.1 -6.21 0.03 0.06 -0.06 -0.03 -33
j End 9.2 -6.06 0.03 0.06 -0.06 -0.03 -31
M13 I End 9.1 -6.06 0.03 0.06 -0.06 -0.03 -31
j End 9.2 -5.18 0.03 0.05 -0.05 -0.02 -22
M14 I End 9.1 -5.18 0.03 0.05 -0.05 -0.02 -22
j End 9.2 -3.76 0.03 0.04 -0.04 -0.01 -7
M15 I End 9.2 -3.76 0.03 0.04 -0.04 -0.01 -7
j End 9.2 -1.96 0.03 0.02 -0.02 0.01 12
M16 I End 9.3 -1.96 0.03 0.02 -0.02 0.01 12
j End 9.3 0.23 0.03 0.00 0.00 0.03 35
M17 I End 9.6 0.23 0.03 0.00 0.00 0.04 36
j End 9.8 1.62 0.03 -0.02 0.02 0.05 51
M18 I End 9.9 1.62 0.03 -0.02 0.02 0.05 51
j End 10.0 3.57 0.03 -0.04 0.04 0.07 72
M19 I End 10.4 3.57 0.04 -0.04 0.04 0.07 73
j End 10.6 4.71 0.04 -0.05 0.05 0.09 86
M20 I End 10.9 4.71 0.04 -0.05 0.05 0.09 87
j End 11.2 5.08 0.04 -0.05 0.05 0.09 92
M21 I End 11.4 5.08 0.04 -0.05 0.05 0.09 93
j End 11.9 2.09 0.04 -0.02 0.02 0.06 63
M22 I End 12.3 2.09 0.04 -0.02 0.02 0.06 65
j End 13.0 -3.17 0.05 0.03 -0.03 0.01 12
53


Figure 6.29. The location of truck 1.
Figure 6.30. Axial force (kips) due to load combination of dead load and live load for truck
in position 1.
54


Figure 6.31. Moment (Kips-ft) due to load combination of daed load and live load for truck
in position 1.
6.4.1.2 Load combination with truck in position 2,
Description: (Self-Weight + Fill+ Truck 2+ Pressure 1+Pressure2)
Table 6.11. Total stress due to load combination with truck in position 2
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress(ksi) Total Stress (psi)
Ml I End 13.2 -2.15 0.05 0.02 -0.02 0.02 23
j End 12.5 2.58 0.04 -0.03 0.03 0.07 70
M2 I End 12.1 2.58 0.04 -0.03 0.03 0.07 69
j End 11.6 5.11 0.04 -0.05 0.05 0.09 93
M3 I End 11.3 5.11 0.04 -0.05 0.05 0.09 92
j End 11.0 4.35 0.04 -0.05 0.05 0.08 83
M4 I End 10.6 4.35 0.04 -0.05 0.05 0.08 82
j End 10.4 2.86 0.04 -0.03 0.03 0.07 66
M5 I End 10.0 2.86 0.03 -0.03 0.03 0.06 64
j End 9.9 0.57 0.03 -0.01 0.01 0.04 40
M6 I End 9.7 0.57 0.03 -0.01 0.01 0.04 40
j End 9.6 -1.09 0.03 0.01 -0.01 0.02 22
M7 I End 9.2 -1.09 0.03 0.01 -0.01 0.02 21
j End 9.1 -3.27 0.03 0.03 -0.03 0.00 -2
55


M8 I End 9.1 -3.27 0.03 0.03 -0.03 0.00 -3
j End 9.0 -4.76 0.03 0.05 -0.05 -0.02 -18
M9 I End 9.0 -4.76 0.03 0.05 -0.05 -0.02 -18
j End 8.9 -5.54 0.03 0.06 -0.06 -0.03 -27
M10 I End 9.0 -5.54 0.03 0.06 -0.06 -0.03 -27
j End 8.9 -5.67 0.03 0.06 -0.06 -0.03 -28
Mil I End 9.0 -5.67 0.03 0.06 -0.06 -0.03 -28
j End 9.0 -5.19 0.03 0.05 -0.05 -0.02 -23
M12 I End 9.0 -5.19 0.03 0.05 -0.05 -0.02 -23
j End 9.1 -4.73 0.03 0.05 -0.05 -0.02 -18
M13 I End 9.0 -4.73 0.03 0.05 -0.05 -0.02 -18
j End 9.0 -3.93 0.03 0.04 -0.04 -0.01 -10
M14 I End 9.0 -3.93 0.03 0.04 -0.04 -0.01 -10
j End 9.0 -2.78 0.03 0.03 -0.03 0.00 2
M15 I End 9.0 -2.78 0.03 0.03 -0.03 0.00 2
j End 9.1 -1.23 0.03 0.01 -0.01 0.02 19
M16 I End 9.1 -1.23 0.03 0.01 -0.01 0.02 19
j End 9.2 0.71 0.03 -0.01 0.01 0.04 39
M17 I End 9.4 0.71 0.03 -0.01 0.01 0.04 40
j End 9.5 1.86 0.03 -0.02 0.02 0.05 53
M18 I End 9.6 1.86 0.03 -0.02 0.02 0.05 53
j End 9.8 3.59 0.03 -0.04 0.04 0.07 71
M19 I End 10.1 3.59 0.04 -0.04 0.04 0.07 72
j End 10.3 4.53 0.04 -0.05 0.05 0.08 83
M20 I End 10.6 4.53 0.04 -0.05 0.05 0.08 84
j End 10.9 4.73 0.04 -0.05 0.05 0.09 87
M21 I End 11.1 4.73 0.04 -0.05 0.05 0.09 88
j End 11.6 1.64 0.04 -0.02 0.02 0.06 57
M22 I End 12.0 1.64 0.04 -0.02 0.02 0.06 59
j End 12.7 -3.63 0.04 0.04 -0.04 0.01 6
56


Figure 6.32. The location of truck 2.
Figure 6.33. Axial force (kips) due to to load combination of daed load and live load for
truck in position 2.
57


Figure 6.34. Moment (kips-ft) due to to load combination of daed load and live load for truck
in position 2.
6.4.1.3 Load combination with truck in position 3,
Description: (Self-Weight + Fill+ Truck 3+ Pressure 1+Pressure2)
Table 6.12. Total stress due to load combination with truck in position 3.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress (ksi) Total Stress (psi)
Ml I End 15.2 0.63 0.05 -0.01 0.01 0.06 59
j End 14.5 3.35 0.05 -0.03 0.03 0.09 85
M2 I End 14.0 3.35 0.05 -0.03 0.03 0.08 83
j End 12.8 4.26 0.04 -0.04 0.04 0.09 89
M3 I End 12.2 4.26 0.04 -0.04 0.04 0.09 87
j End 11.5 2.58 0.04 -0.03 0.03 0.07 67
M4 I End 11.1 2.58 0.04 -0.03 0.03 0.07 65
j End 10.6 0.75 0.04 -0.01 0.01 0.04 44
M5 I End 10.1 0.75 0.04 -0.01 0.01 0.04 43
j End 9.8 -1.35 0.03 0.01 -0.01 0.02 20
M6 I End 9.7 -1.35 0.03 0.01 -0.01 0.02 20
j End 9.4 -2.35 0.03 0.02 -0.02 0.01 8
M7 I End 9.2 -2.35 0.03 0.02 -0.02 0.01 7
58


j End 9.0 -3.64 0.03 0.04 -0.04 -0.01 -7
M8 I End 9.0 -3.64 0.03 0.04 -0.04 -0.01 -7
j End 9.0 -4.31 0.03 0.04 -0.04 -0.01 -14
M9 I End 9.0 -4.31 0.03 0.04 -0.04 -0.01 -14
j End 9.0 -4.59 0.03 0.05 -0.05 -0.02 -17
M10 I End 9.0 -4.59 0.03 0.05 -0.05 -0.02 -16
j End 9.0 -4.51 0.03 0.05 -0.05 -0.02 -16
Mil I End 9.1 -4.51 0.03 0.05 -0.05 -0.02 -15
j End 9.1 -4.10 0.03 0.04 -0.04 -0.01 -11
M12 I End 9.1 -4.10 0.03 0.04 -0.04 -0.01 -11
j End 9.1 -3.77 0.03 0.04 -0.04 -0.01 -7
M13 I End 9.1 -3.77 0.03 0.04 -0.04 -0.01 -8
j End 9.1 -3.09 0.03 0.03 -0.03 0.00 -1
M14 I End 9.1 -3.09 0.03 0.03 -0.03 0.00 -1
j End 9.1 -2.07 0.03 0.02 -0.02 0.01 10
M15 I End 9.1 -2.07 0.03 0.02 -0.02 0.01 10
j End 9.1 -0.65 0.03 0.01 -0.01 0.02 25
M16 I End 9.2 -0.65 0.03 0.01 -0.01 0.03 25
j End 9.2 1.15 0.03 -0.01 0.01 0.04 44
M17 I End 9.5 1.15 0.03 -0.01 0.01 0.04 45
j End 9.6 2.15 0.03 -0.02 0.02 0.06 56
M18 I End 9.7 2.15 0.03 -0.02 0.02 0.06 56
j End 9.8 3.72 0.03 -0.04 0.04 0.07 73
M19 I End 10.1 3.72 0.04 -0.04 0.04 0.07 74
j End 10.3 4.48 0.04 -0.05 0.05 0.08 83
M20 I End 10.6 4.48 0.04 -0.05 0.05 0.08 84
j End 10.9 4.49 0.04 -0.05 0.05 0.08 85
M21 I End 11.1 4.49 0.04 -0.05 0.05 0.09 85
j End 11.6 1.14 0.04 -0.01 0.01 0.05 52
M22 I End 11.9 1.14 0.04 -0.01 0.01 0.05 53
j End 12.6 -4.46 0.04 0.05 -0.05 0.00 -3
59


Figure 6.35. The location of truck 3.
Figure 6.36. Axial force (kips) due to to load combination of daed load and live load for
truck in position 3.
60


Figure 6.37. Moment (kips-ft) due to to load combination of daed load and live load for truck
in position 3.
6.4.1.4 Load combination with truck in position 4,
Description: (Self-Weight + Fill+ Truck 3+ Pressure 1+Pressure2)
Table 6.13. Final results due to load combination with truck in position 4.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Total Stress Stress (ksi) (psi)
Ml I End 13.4 -0.86 0.05 0.01 -0.01 0.04 37
j End 12.3 2.23 0.04 -0.02 0.02 0.07 66
M2 I End 11.8 2.23 0.04 -0.02 0.02 0.06 64
j End 11.0 3.91 0.04 -0.04 0.04 0.08 79
M3 I End 10.6 3.91 0.04 -0.04 0.04 0.08 78
j End 10.1 3.01 0.04 -0.03 0.03 0.07 67
M4 I End 9.8 3.01 0.03 -0.03 0.03 0.07 65
j End 9.4 1.79 0.03 -0.02 0.02 0.05 51
M5 I End 9.1 1.79 0.03 -0.02 0.02 0.05 50
j End 8.9 0.14 0.03 0.00 0.00 0.03 32
M6 I End 8.8 0.14 0.03 0.00 0.00 0.03 32
j End 8.6 -0.72 0.03 0.01 -0.01 0.02 22
M7 I End 8.4 -0.72 0.03 0.01 -0.01 0.02 22
j End 8.3 -2.21 0.03 0.02 -0.02 0.01 6
61


M8 I End 8.3 -2.21 0.03 0.02 -0.02 0.01 6
j End 8.3 -3.29 0.03 0.03 -0.03 -0.01 -6
M9 I End 8.3 -3.29 0.03 0.03 -0.03 -0.01 -5
j End 8.3 -3.95 0.03 0.04 -0.04 -0.01 -12
M10 I End 8.3 -3.95 0.03 0.04 -0.04 -0.01 -12
j End 8.3 -4.23 0.03 0.04 -0.04 -0.02 -15
Mil I End 8.4 -4.23 0.03 0.04 -0.04 -0.01 -15
j End 8.4 -4.13 0.03 0.04 -0.04 -0.01 -14
M12 I End 8.4 -4.13 0.03 0.04 -0.04 -0.01 -14
j End 8.4 -4.05 0.03 0.04 -0.04 -0.01 -13
M13 I End 8.3 -4.05 0.03 0.04 -0.04 -0.01 -13
j End 8.3 -3.60 0.03 0.04 -0.04 -0.01 -9
M14 I End 8.3 -3.60 0.03 0.04 -0.04 -0.01 -9
j End 8.3 -2.77 0.03 0.03 -0.03 0.00 0
M15 I End 8.3 -2.77 0.03 0.03 -0.03 0.00 0
j End 8.4 -1.52 0.03 0.02 -0.02 0.01 13
M16 I End 8.4 -1.52 0.03 0.02 -0.02 0.01 13
j End 8.4 0.15 0.03 0.00 0.00 0.03 31
M17 I End 8.7 0.15 0.03 0.00 0.00 0.03 32
j End 8.8 1.17 0.03 -0.01 0.01 0.04 43
M18 I End 8.9 1.17 0.03 -0.01 0.01 0.04 43
j End 9.0 2.78 0.03 -0.03 0.03 0.06 60
M19 I End 9.3 2.78 0.03 -0.03 0.03 0.06 61
j End 9.5 3.74 0.03 -0.04 0.04 0.07 72
M20 I End 9.9 3.74 0.03 -0.04 0.04 0.07 73
j End 10.2 4.13 0.04 -0.04 0.04 0.08 78
M21 I End 10.4 4.13 0.04 -0.04 0.04 0.08 79
j End 10.9 1.62 0.04 -0.02 0.02 0.05 55
M22 I End 11.4 1.62 0.04 -0.02 0.02 0.06 56
j End 12.0 -2.70 0.04 0.03 -0.03 0.01 14
62


Figure 6.38. The location of truck 4.
Figure 6.39. Axial force (kips) due to to load combination of daed load and live load for
truck in position 4.
63


Figure 6.40. Moment (kips-ft) due to to load combination of daed load and live load for truck
in position 4.
6.4.2 Risa 2D Frame with three hinges.
When the bending moment in the cross-section results in significant net tension, the joints between arch stones will tend to separate. We can approximate this behavior by adding hinges at high tension locations. Risa 2D Frame analysis of the collapse load of the frame leads to high tension in three different locations. Therefore, it is important when solving these problems using simple methods to permit joint rotation to simulate stone masonry joint separations in the real arch.
Consequently, hinge forms were assumed in different three positions at joint releases adjacent to N4, N12, and N20 as shown in Figure 6.41:
64


Figure 6.41. Model of the arch with hinges in Risa 2D Frame.
These hinges leads to reduce moment value and total stress on the members as presents below for all load combination:
6.4.2.1 Load combination with truck in position 1 Table 6.14. Total stress due to load combination with truck in position 1 with three hinges.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bot Bending Bending (ksi) (ksi) Total Stress(ksi) Total Stress (psi)
Ml I End 15.21 -31.01 0.05 0.32 -0.32 -0.27 -270
j End 14.54 -15.05 0.05 0.16 -0.16 -0.11 -106
M2 I End 14.74 -15.05 0.05 0.16 -0.16 -0.11 -106
j End 14.26 -4.25 0.05 0.04 -0.04 0.01 5
M3 I End 15.01 -4.25 0.05 0.04 -0.04 0.01 8
j End 14.72 0.00 0.05 0.00 0.00 0.05 51
M4 I End 14.86 0.00 0.05 0.00 0.00 0.05 52
j End 14.65 2.26 0.05 -0.02 0.02 0.07 74
M5 I End 14.71 2.26 0.05 -0.02 0.02 0.07 75
j End 14.57 2.56 0.05 -0.03 0.03 0.08 77
M6 I End 14.53 2.56 0.05 -0.03 0.03 0.08 77
j End 14.41 3.23 0.05 -0.03 0.03 0.08 84
65


M7 I End 14.36 3.23 0.05 -0.03 0.03 0.08 84
j End 14.30 2.12 0.05 -0.02 0.02 0.07 72
M8 I End 14.30 2.12 0.05 -0.02 0.02 0.07 72
j End 14.25 1.16 0.05 -0.01 0.01 0.06 62
M9 I End 14.28 1.16 0.05 -0.01 0.01 0.06 62
j End 14.24 0.39 0.05 0.00 0.00 0.05 53
M10 I End 14.28 0.39 0.05 0.00 0.00 0.05 54
j End 14.22 -0.06 0.05 0.00 0.00 0.05 49
Mil I End 14.31 -0.06 0.05 0.00 0.00 0.05 49
j End 14.28 0.00 0.05 0.00 0.00 0.05 50
M12 I End 14.28 0.00 0.05 0.00 0.00 0.05 50
j End 14.31 -0.06 0.05 0.00 0.00 0.05 49
M13 I End 14.22 -0.06 0.05 0.00 0.00 0.05 49
j End 14.28 0.39 0.05 0.00 0.00 0.05 54
M14 I End 14.24 0.39 0.05 0.00 0.00 0.05 53
j End 14.28 1.16 0.05 -0.01 0.01 0.06 62
M15 I End 14.25 1.16 0.05 -0.01 0.01 0.06 62
j End 14.30 2.12 0.05 -0.02 0.02 0.07 72
M16 I End 14.30 2.12 0.05 -0.02 0.02 0.07 72
j End 14.36 3.23 0.05 -0.03 0.03 0.08 84
M17 I End 14.41 3.23 0.05 -0.03 0.03 0.08 84
j End 14.53 2.56 0.05 -0.03 0.03 0.08 77
M18 I End 14.57 2.56 0.05 -0.03 0.03 0.08 77
j End 14.71 2.26 0.05 -0.02 0.02 0.07 75
M19 I End 14.65 2.26 0.05 -0.02 0.02 0.07 74
j End 14.86 0.00 0.05 0.00 0.00 0.05 52
M20 I End 14.72 0.00 0.05 0.00 0.00 0.05 51
j End 15.01 -4.25 0.05 0.04 -0.04 0.01 8
M21 I End 14.26 -4.25 0.05 0.04 -0.04 0.01 5
j End 14.74 -15.05 0.05 0.16 -0.16 -0.11 -106
M22 I End 14.54 -15.05 0.05 0.16 -0.16 -0.11 -106
j End 15.21 -31.01 0.05 0.32 -0.32 -0.27 -270
66


Figure 6.42. Axial force (kips) due to load combination of dead load and live load for truck
in position 1 with three hinges.
Figure 6.43. Moment (kips-ft) due to load combination of dead load and live load for truck in
position 1 with three hinges
67


6.4.2.2 Load combination with truck in position 2
Table 6.15. Total stress due to load combination with truck in position 2 with three hinges.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress(ksi) Total Stress (psi)
Ml I End 15.15 -26.84 0.05 0.28 -0.28 -0.23 -227
j End 14.48 -12.70 0.05 0.13 -0.13 -0.08 -82
M2 I End 14.58 -12.70 0.05 0.13 -0.13 -0.08 -82
j End 14.09 -3.31 0.05 0.03 -0.03 0.01 15
M3 I End 14.65 -3.31 0.05 0.03 -0.03 0.02 16
j End 14.37 0.00 0.05 0.00 0.00 0.05 50
M4 I End 14.41 0.00 0.05 0.00 0.00 0.05 50
j End 14.19 1.48 0.05 -0.02 0.02 0.06 65
M5 I End 14.16 1.48 0.05 -0.02 0.02 0.06 65
j End 14.02 1.18 0.05 -0.01 0.01 0.06 61
M6 I End 13.96 1.18 0.05 -0.01 0.01 0.06 61
j End 13.77 1.32 0.05 -0.01 0.01 0.06 62
M7 I End 13.67 1.32 0.05 -0.01 0.01 0.06 61
j End 13.54 0.08 0.05 0.00 0.00 0.05 48
M8 I End 13.55 0.08 0.05 0.00 0.00 0.05 48
j End 13.44 -0.67 0.05 0.01 -0.01 0.04 40
M9 I End 13.49 -0.67 0.05 0.01 -0.01 0.04 40
j End 13.41 -0.90 0.05 0.01 -0.01 0.04 37
M10 I End 13.48 -0.90 0.05 0.01 -0.01 0.04 37
j End 13.43 -0.66 0.05 0.01 -0.01 0.04 40
Mil I End 13.54 -0.66 0.05 0.01 -0.01 0.04 40
j End 13.52 0.00 0.05 0.00 0.00 0.05 47
M12 I End 13.56 0.00 0.05 0.00 0.00 0.05 47
j End 13.58 0.26 0.05 0.00 0.00 0.05 50
M13 I End 13.50 0.26 0.05 0.00 0.00 0.05 50
j End 13.53 0.67 0.05 -0.01 0.01 0.05 54
M14 I End 13.47 0.67 0.05 -0.01 0.01 0.05 54
j End 13.51 1.24 0.05 -0.01 0.01 0.06 60
M15 I End 13.47 1.24 0.05 -0.01 0.01 0.06 60
j End 13.53 2.03 0.05 -0.02 0.02 0.07 68
M16 I End 13.52 2.03 0.05 -0.02 0.02 0.07 68
j End 13.59 3.01 0.05 -0.03 0.03 0.08 78
M17 I End 13.62 3.01 0.05 -0.03 0.03 0.08 79
j End 13.74 2.34 0.05 -0.02 0.02 0.07 72
68


M18 I End 13.78 2.34 0.05 -0.02 0.02 0.07 72
j End 13.92 2.06 0.05 -0.02 0.02 0.07 70
M19 I End 13.88 2.06 0.05 -0.02 0.02 0.07 70
j End 14.09 0.00 0.05 0.00 0.00 0.05 49
M20 I End 13.98 0.00 0.05 0.00 0.00 0.05 49
j End 14.27 -3.88 0.05 0.04 -0.04 0.01 9
M21 I End 13.61 -3.88 0.05 0.04 -0.04 0.01 7
j End 14.10 -13.85 0.05 0.14 -0.14 -0.10 -95
M22 I End 13.96 -13.85 0.05 0.14 -0.14 -0.10 -96
j End 14.63 -28.58 0.05 0.30 -0.30 -0.25 -247
Figure 6.44. Axial force (kips) due to load combination of dead load and live load for truck
in position 2 with three hinges.
69


tL.
Figure 6.45. Moment (kips-ft) due to load combination of dead load and live load for truck in
position 2 with three hinges.
6.4.2.3 Load combination with truck in position 3 Table 6.16. Total stress due to load combination with truck in position 3 with three hinges.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress (ksi) Total Stress (psi)
Ml I End 16.87 -17.75 0.06 0.18 -0.18 -0.13 -126
j End 16.20 -7.67 0.06 0.08 -0.08 -0.02 -24
M2 I End 16.05 -7.67 0.06 0.08 -0.08 -0.02 -24
j End 14.83 -1.43 0.05 0.01 -0.01 0.04 37
M3 I End 14.99 -1.43 0.05 0.01 -0.01 0.04 37
j End 14.27 0.00 0.05 0.00 0.00 0.05 50
M4 I End 14.14 0.00 0.05 0.00 0.00 0.05 49
j End 13.61 0.42 0.05 0.00 0.00 0.05 52
M5 I End 13.48 0.42 0.05 0.00 0.00 0.05 51
j End 13.12 -0.21 0.05 0.00 0.00 0.04 43
M6 I End 13.07 -0.21 0.05 0.00 0.00 0.04 43
j End 12.75 0.11 0.04 0.00 0.00 0.05 45
M7 I End 12.71 0.11 0.04 0.00 0.00 0.05 45
j End 12.55 -0.55 0.04 0.01 -0.01 0.04 38
70


M8 I End 12.58 -0.55 0.04 0.01 -0.01 0.04 38
j End 12.53 -0.75 0.04 0.01 -0.01 0.04 36
M9 I End 12.59 -0.75 0.04 0.01 -0.01 0.04 36
j End 12.55 -0.69 0.04 0.01 -0.01 0.04 36
M10 I End 12.63 -0.69 0.04 0.01 -0.01 0.04 37
j End 12.60 -0.44 0.04 0.00 0.00 0.04 39
Mil I End 12.71 -0.44 0.04 0.00 0.00 0.04 40
j End 12.69 0.00 0.04 0.00 0.00 0.04 44
M12 I End 12.71 0.00 0.04 0.00 0.00 0.04 44
j End 12.73 0.07 0.04 0.00 0.00 0.04 45
M13 I End 12.64 0.07 0.04 0.00 0.00 0.04 45
j End 12.67 0.33 0.04 0.00 0.00 0.05 47
M14 I End 12.61 0.33 0.04 0.00 0.00 0.05 47
j End 12.65 0.78 0.04 -0.01 0.01 0.05 52
M15 I End 12.61 0.78 0.04 -0.01 0.01 0.05 52
j End 12.66 1.49 0.04 -0.02 0.02 0.06 59
M16 I End 12.65 1.49 0.04 -0.02 0.02 0.06 59
j End 12.72 2.42 0.04 -0.03 0.03 0.07 69
M17 I End 12.75 2.42 0.04 -0.03 0.03 0.07 69
j End 12.88 1.86 0.04 -0.02 0.02 0.06 64
M18 I End 12.92 1.86 0.04 -0.02 0.02 0.06 64
j End 13.07 1.73 0.05 -0.02 0.02 0.06 63
M19 I End 13.05 1.73 0.05 -0.02 0.02 0.06 63
j End 13.26 0.00 0.05 0.00 0.00 0.05 46
M20 I End 13.21 0.00 0.05 0.00 0.00 0.05 46
j End 13.49 -3.35 0.05 0.03 -0.03 0.01 12
M21 I End 12.96 -3.35 0.05 0.03 -0.03 0.01 10
j End 13.45 -12.27 0.05 0.13 -0.13 -0.08 -81
M22 I End 13.40 -12.27 0.05 0.13 -0.13 -0.08 -81
j End 14.06 -25.47 0.05 0.27 -0.27 -0.22 -216
71


Figure 6.46. Axial force (kips) due to load combination of dead load and live load for truck
in position 3 with three hinges.
Figure 6.47. Moment (kips-ft) due to load combination of dead load and live load for truck in
position 3 with three hinges.
72


6.4.2.4 Load combination with truck in position 4
Table 6.17. Total stress due to load combination with truck in position 4 with three hinges.
Member Label Section Axial [k] Moment [k-ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress (ksi) Total Stress (psi)
Ml I End 14.92 -19.64 0.05 0.20 -0.20 -0.15 -153
j End 13.84 -9.23 0.05 0.10 -0.10 -0.05 -48
M2 I End 13.74 -9.23 0.05 0.10 -0.10 -0.05 -48
j End 12.96 -2.24 0.04 0.02 -0.02 0.02 22
M3 I End 13.25 -2.24 0.05 0.02 -0.02 0.02 23
j End 12.78 0.00 0.04 0.00 0.00 0.04 44
M4 I End 12.73 0.00 0.04 0.00 0.00 0.04 44
j End 12.39 1.06 0.04 -0.01 0.01 0.05 54
M5 I End 12.34 1.06 0.04 -0.01 0.01 0.05 54
j End 12.11 0.93 0.04 -0.01 0.01 0.05 52
M6 I End 12.06 0.93 0.04 -0.01 0.01 0.05 52
j End 11.90 1.43 0.04 -0.01 0.01 0.06 56
M7 I End 11.87 1.43 0.04 -0.01 0.01 0.06 56
j End 11.80 0.64 0.04 -0.01 0.01 0.05 48
M8 I End 11.82 0.64 0.04 -0.01 0.01 0.05 48
j End 11.77 0.09 0.04 0.00 0.00 0.04 42
M9 I End 11.81 0.09 0.04 0.00 0.00 0.04 42
j End 11.77 -0.17 0.04 0.00 0.00 0.04 39
M10 I End 11.84 -0.17 0.04 0.00 0.00 0.04 39
j End 11.82 -0.19 0.04 0.00 0.00 0.04 39
Mil I End 11.91 -0.19 0.04 0.00 0.00 0.04 39
j End 11.90 0.00 0.04 0.00 0.00 0.04 41
M12 I End 11.90 0.00 0.04 0.00 0.00 0.04 41
j End 11.92 -0.11 0.04 0.00 0.00 0.04 40
M13 I End 11.82 -0.11 0.04 0.00 0.00 0.04 40
j End 11.85 0.00 0.04 0.00 0.00 0.04 41
M14 I End 11.78 0.00 0.04 0.00 0.00 0.04 41
j End 11.82 0.34 0.04 0.00 0.00 0.04 45
M15 I End 11.78 0.34 0.04 0.00 0.00 0.04 44
j End 11.83 0.97 0.04 -0.01 0.01 0.05 51
M16 I End 11.82 0.97 0.04 -0.01 0.01 0.05 51
j End 11.89 1.85 0.04 -0.02 0.02 0.06 61
M17 I End 11.93 1.85 0.04 -0.02 0.02 0.06 61
j End 12.05 1.40 0.04 -0.01 0.01 0.06 56
M18 I End 12.10 1.40 0.04 -0.01 0.01 0.06 57
73


j End 12.24 1.41 0.04 -0.01 0.01 0.06 57
M19 I End 12.26 1.41 0.04 -0.01 0.01 0.06 57
j End 12.47 0.00 0.04 0.00 0.00 0.04 43
M20 I End 12.46 0.00 0.04 0.00 0.00 0.04 43
j End 12.75 -2.84 0.04 0.03 -0.03 0.01 15
M21 I End 12.34 -2.84 0.04 0.03 -0.03 0.01 13
j End 12.83 -10.75 0.04 0.11 -0.11 -0.07 -67
M22 I End 12.85 -10.75 0.04 0.11 -0.11 -0.07 -67
j End 13.52 -22.49 0.05 0.23 -0.23 -0.19 -187
Figure 6.48. Axial force (kips) due to load combination of dead load and live load for truck
in position 4 with three hinges.
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Figure 6.49. Moment(kips-ft) due to load combination of dead load and live load for truck in
position 4 with three hinges.
6.5 Analysis Using Risa 2D Finite Element.
This method is based on Finite Element. The arch is modelled as a many plates which have the same properties. Also, in this analysis, fixed supports are applied on the abutments. Figure 6.50 shows the model of the arch using Risa Finite Element.
Figure 6.50. Finit element models.
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6.5.1 Loads.
Table 6.18. Dead load values in Risa FE.
Node# Self-Wight Load (k/ft) Fill Load (k/ft) Pressure Load Pressure Load (Right) (k/ft) (Left) (k/ft)
N1 0.32 1.17 0.351
N2 0.32 0.92004 0.276012
N3 0.32 0.738 0.2214
N4 0.32 0.63 0.189
N5 0.32 0.5508 0.16524
N6 0.32 0.498 0.1494
N7 0.32 0.45 0.135
N8 0.32 0.4248 0.12744
N9 0.32 0.405 0.1215
N10 0.32 0.39 0.117
Nil 0.32 0.38004 0.114012
N12 0.32 0.375 0.1125
N13 0.32 0.38004 0.114012
N14 0.32 0.39 0.117
N15 0.32 0.405 0.1215
N16 0.32 0.4248 0.12744
N17 0.32 0.45 0.135
N18 0.32 0.498 0.1494
N19 0.32 0.5508 0.16524
N20 0.32 0.63 0.189
N21 0.32 0.738 0.2214
N22 0.32 0.92004 0.276012
N23 0.32 1.17 0.351
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Figure 6.51. Self-weight load Fill load in Risa FE.
Figure 6.52. Pressure load on the right side of the arch in Risa FE.
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Figure 6.53. Pressure load on the left side of the arch in Risa FE.
The Ely Stone Bridge was applied the dead load and live load with four different load combustion based on four positions. The description of the position of live load and the result for dead load and live load is shown below:
Figure 6.54. Truck 1 position in Risa FE.
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Figure 6.55. Truck 2 position in Risa FE.
Figure 6.56. Truck 3 position in Risa FE.
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-26k
-26k
Figure 6.57. Truck 4 position in Risa FE.
Results for LC14, Self-Wight+Fill+Pressure 1+Pressure 2
Figure 6.58. The stress by dead load in Risa FE.
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| Results for LC 9, Self-Wight+FilETruck 1 ♦Pressure 1+Pressuj
Figure 6.59. The stress by dead load and live load “truck 1” in Risa FE.
Figure 6.60. The stress by dead load and live load “truck 2” in Risa FE.
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Results for LC11, Self-Wight+Fill+Truck 3+Pressure 1+Pressj
Figure 6.61. The stress by dead load and live load “truck 3” in Risa FE.
Results for LC 12, Self-Wight+Fill+Truck 4+Pressure 1+Press]
Figure 6.62. The stress by dead load and live load “truck 4” in Risa FE.
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CHAPTER VII
7 DISCUSSION
7.1 Introduction to Comparison of Different Calculation Methods
There are many empirical and numerical methods that help to determine the failure load with 2D and 3D modelling. Use of these methods helps us to get more understanding of the behavior of the structure under multiple loads. Using these kinds of methods brings many, but not all effects and circumstances into consideration. However, it is necessary to spend much time in creating models.
7.1.1 The MEXE and Ring
A comparative calculation was done for the same bridge by using two different methods which are the MEXE and Ring methods. A single-axle was located over the span and then the ultimate load calculate was calculated and the results shown in Table 7.1.
Table 7.1. Results from MEXE method and Ring.
Method Permissible axle load
MEXE Method 120 kips
Ring 3.2 Method 220 kips
The result from the MEXE Modified method, an empirical method, is based in part on the engineer’s discretion and judgement with regard to condition. The MEXE method can be a helpful method to estimate the load capacity for masonry arch bridges. On the other hand, the result from Ring is based on the attributes of the bridge such as fill angle of internal friction, friction coefficient between stones, and unit weight for the material. Some of parameters were assumed because it is impossible to measure them in an existing bridge, i.e.
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friction coefficients between the stones of the arch barrel and compressive strength of the
masonry.
From the comparison results on the Table above, the MEXE method gives a permissible axle load of 120 kip. However, Ring analysis gave a permissible axle load 220 kip for a single axle. Hence, the Ring method gives a permissible axle load about 2.0 times the permissible axle load from the MEXE method. The biggest reason for this difference in the load factor value is that Ring software takes into consideration arch failure and already applied hinges on those locations that have a mechanism failure which can lead to collapse, whereas MEXE is a more simplistic and inherently conservative approach.
7.1.2 Risa 2D Frame and Ring.
Different comparative calculations were done here between Risa 2D Frame and Ring. In this comparison, the design load factor for the Ely Stone Bridge using the Ring method was applied as a live load on the middle of the strip of arch barrel using the Risa 2D Frame and Ring software as shown in the Figure below:
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Figure 7.2. Model of the Ely Stone Bridge in Ring.
The results for the Ely Stone Bridge from two methods match better than those for the MEXE and Ring methods. Where the analysis for both methods gives the same critical locations, which have tension area as seen in the axial force diagram and the moment diagram Figures below:
Figure 7.3. Axial force (kips) diagram based on Ring’s 220 kips axle load, in Risa 2D.
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Figure 7.4. Moment diagram by the critical design load in Risa 2D.
Figure 7.5. Failure mechanism for the The Ely Stone Bridge from RING2.0.
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Figure 7.6. Axial force diagram by the critical design load in Ring.
Figure 7.7. Moment diagram by the critical design load in Ring.
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Full Text

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COMPARISON OF STRUCTURAL ANALYS I S METHODS FOR MASONRY ARCHES by ASRAR AHMED EL TOMI B.S., University of Tripoli, 2007 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering Program 201 7

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ii This thesis for the Master of Science degree by Asrar Ahmed El Tomi has been approved for the Civil Engineering Program by Kevin Rens , Chair Frederick Rutz , Advisor Carnot Nogueira Date May 13 , 2017

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iii El Tomi , Asrar Ahmed (M.S., Civil Engineering Program ) C omparison of Structural Analys i s Methods for Masonry A rches Thesis directed by Associate Professor Frederick Rutz ABSTRACT Different methods for the structural analysis for a m asonry arch bridge are examined under similar conditions and the results of these analyses are compared. The MEXE method, often used in the United Kingdom, is compared to the method utilized by Ring, a software specifically intended for masonry arch bridges. The Ring method indicates approximately double the structur al capacity as that from the MEXE method. Ring is also compared to the results of a nalyses from the Risa 2D Frame and Risa 2D Finite Element methods. The methods give different results, with Ring and MEXE method resulting in an allowable axle load. However, Risa 2D Frame and Risa 2D Finite Element lead to stresses, so are indirectly related to MEXE and Ring. The Frame and Finite Element methods proved less satisfactory than MEXE or Ring because of analytical difficulties in modelling the non linear response of joint separ ations between individual stone units within the arch. The form and content of this abstract are approved. I recommend its publication. Approved: Frederick Rutz

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iv ACKNOWLEDGEMENTS My sincerest thanks to my advisor, Dr. Frederick Rutz for guidance, support, and encouragement for two semesters. I would never have been able to finish my thesis without his guidance. Also, thank you Dr. Frederick Rutz for preparing me to write this thesi s. My sincerest thanks to Dr. Kevin Rens for approv ing me to be a part of the Ely Stone Bridge team. Also, thank you to Dr. Kevin Rens and Dr. Carnot Nogueira for being a part of my thesis committee . I would like to express the deepest appreciation to my husband, my little son, family, and friends. Finally, thank you to Department of Engineering in the University of Colorado Denver and all the Structural Faculty in giving me the opportunity to pursue a graduate degree .

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v TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ ........ 1 1.1 Introduction ................................ ................................ ................................ ................ 1 1.2 Research Objective ................................ ................................ ................................ ..... 1 1.3 Outline of This Thesis ................................ ................................ ................................ 1 II. LITERATURE REVIEW ................................ ................................ ............................. 3 2.1 Introduction ................................ ................................ ................................ ................ 3 2.2 History of Stone Masonry Arch Bridge ................................ ................................ ..... 3 2.3 Arches ................................ ................................ ................................ ......................... 4 2.3.1 Arches classified. ................................ ................................ ................................ 6 2.3.2 Behavior of the arch under load. ................................ ................................ ......... 8 2.4 Description of the Stone Masonry Arch Bridge ................................ ......................... 8 2.4.1 Materials. ................................ ................................ ................................ ............ 9 2.5 Structural Analysis of Old Masonry Arch Bridge ................................ ...................... 9 III. THE ELY STONE BRIDGE ................................ ................................ ...................... 11 3.1 Intr oduction ................................ ................................ ................................ .............. 11 3.2 Background and Description ................................ ................................ .................... 11 IV. ANALYSIS METHODS: GENERAL ................................ ................................ ........ 15 4.1 Introduction ................................ ................................ ................................ .............. 15 4.2 Methods Used for Modeling ................................ ................................ .................... 15 4.2.1 Three Hinged Arch method ................................ ................................ .............. 16 4.2.2 The Modified MEXE method ................................ ................................ ........... 17 4.2.3 RING 3.2 ................................ ................................ ................................ ........... 18 4.2.4 RISA ................................ ................................ ................................ ................. 19 4.2.5 Three dimensional FEM ................................ ................................ ................... 19 V. LOADS ................................ ................................ ................................ ....................... 20 5.1 Introduction ................................ ................................ ................................ .............. 20 5.2 Design Codes and Regulations ................................ ................................ ................. 20 5.3 Loads ................................ ................................ ................................ ........................ 20 5.3.1 Dead load. ................................ ................................ ................................ ......... 21

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vi 5.3.2 ................................ ................................ ....................... 22 5.4 Load Combinations ................................ ................................ ................................ .. 24 5.4.1 Combined dead load and live loads ................................ ................................ .. 24 VI. ANALYSIS AND COMPARSION OF THE RESULTS ................................ ........... 29 6.1 Linearly Elastic Analysis ................................ ................................ ......................... 29 6.2 Analysis Using MEXE ................................ ................................ ............................. 34 6.2.1 Calculate the provisional axle load. ................................ ................................ .. 36 6.2.1.1 Calculate modification factors. ................................ ................................ .. 38 6.2.2 Modified axle load. ................................ ................................ ........................... 42 6.3 Analysis Using Ring. ................................ ................................ ................................ 43 6.4 Analysis Using Risa 2D Frame. ................................ ................................ ............... 47 6.4.1 Loads ................................ ................................ ................................ ................. 48 6.4.1.1 Load com bination with truck in position 1 ................................ ................ 52 6.4.1.2 Load combination with truck in position 2. ................................ ............... 55 6.4.1.3 Load combination with truck in position 3. ................................ ............... 58 6.4.1.4 Load combination with truck in position 4. ................................ ............... 61 6.4.2 Risa 2D Frame with three hinges. ................................ ................................ ..... 64 6.4.2.1 Load combination with truck in position 1 ................................ ................ 65 6.4.2.2 Load combination with truck in position 2 ................................ ................ 68 6.4.2.3 Load combination with truck in position 3 ................................ ................ 70 6.4.2.4 Load combination with truck in position 4 ................................ ................ 73 6.5 Analysis Using Risa 2D Finite Element. ................................ ................................ .. 75 6.5.1 Loads. ................................ ................................ ................................ ................ 76 VII. DISCUSSION ................................ ................................ ................................ ............. 83 7.1 Introduction t o Comparison of Different Calculation Methods ............................... 83 7.1.1 The MEXE and Ring ................................ ................................ ........................ 83 7.1.2 Risa 2D Frame and Ring. ................................ ................................ .................. 84 7.2 Conclusion ................................ ................................ ................................ ................ 89 7.3 Recommenda tions for Further Research ................................ ................................ .. 90 REFERENCES ................................ ................................ ................................ ....................... 91 APPENDIX A . Fi eld Visitation ................................ ................................ ................................ ............. 93

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vii B. S preadsheet for MEXE Methods ................................ ................................ .......... 104 C. R e port from RING 3.2 Analysis ................................ ................................ .......... 106

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viii LIST OF TABLES TABLE 6.1. Barrel factor (The highways agency, 2001). ................................ ................................ ... 40 6.2. Fill factor (The highways agency, 2001). ................................ ................................ ....... 40 6.3. Width factor (The highways agency, 2001). ................................ ................................ .... 41 6.4. Depth factor (The highways agency, 2001). ................................ ................................ .... 41 6.5. Mortar factor (The highwa ys agency, 2001). ................................ ................................ ... 42 6.6. Parameters used in Ring for the Ely Stone Bridge. ................................ ......................... 43 6.7. Load factor used in Ring for the Ely Stone Bridge. ................................ ......................... 43 6.8. Parameters used in Risa 2D Frame for the Ely Stone Bridge. ................................ ........ 48 6.9. Dead load values in Risa 2D Frame. ................................ ................................ ................ 48 6.10. Total stress due to load combination with tr uck in position 1. ................................ ...... 52 6.11. Total stress due to load combination with truck in position 2 ................................ ....... 55 6.12. Total stress due to load combination with truck in position 3. ................................ ...... 58 6.13. Final results due to load combination with truck in position 4. ................................ ..... 61 6.14. Total stre ss due to load combination with truck in position 1 with three hinges. .......... 65 6.15. Total stress due to load combination with truck in position 2 with three hinges. .......... 68 6.16. Total stress due to load combination with truck in position 3 with three hinges. .......... 70 6.17. Total stress due to load combination with truck in position 4 with three hinges. .......... 73 6.18.Dead load values in Risa FE. ................................ ................................ .......................... 76 7.1. Results from MEXE method and Ring. ................................ ................................ .......... 83

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ix LIST OF FIGURES FIGURE 2.1. Distribution forces in the arch. ................................ ................................ ........................... 5 2.2. Corbelled arch. ................................ ................................ ................................ ................... 5 2.3. Fixed fixed arch. ................................ ................................ ................................ ................ 7 2.4. T wo hinged arch. ................................ ................................ ................................ ............... 7 2.5. Three hinged arch. ................................ ................................ ................................ ............. 7 2.6. Important parts of an arch bridge. ................................ ................................ ...................... 8 2.7. Arch and parapet in bridge. ................................ ................................ ................................ 9 3.1 A photograph of the bridge when it was registered as a historic place. ............................ 12 3.2. Satellite view of the Ely Stone Bridge. ................................ ................................ ............ 13 3.3. Side view of the Ely Stone Bridge at east side. ................................ ............................... 13 3.4. Side view of the Ely Stone Bridge at west side. ................................ .............................. 14 4.1. Three hinged arch. ................................ ................................ ................................ ........... 16 4.2. Arch bridge dimensions for MEXE method (Robinson BEng, 2000). ............................ 18 4.3. Shows explanations a RING2.0 analysis (limitState, 2009). ................................ ........... 19 5.1. Illustration of arch barrel strip. ................................ ................................ ........................ 21 5.2. Weight of fill along the strip of arch barrel. ................................ ................................ .... 22 ................................ ................................ ............................ 22 ................................ ................................ ............................ 23 ................................ ................................ ............................ 2 3 ................................ ................................ ............................ 24 5.7. Load combination with truck at position 1. ................................ ................................ ..... 25

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x 5.8. Load com bination with truck at position 2. ................................ ................................ ..... 26 5.9. Load combination with tr uck at position 3. ................................ ................................ ..... 27 5.10. Load combination with truck at position 4. ................................ ................................ ... 28 6.1. Stress of strip of arch barrel due to dead load (self wight, fill). ................................ ...... 29 6.2. Stress of strip of arch barrel due to live load (truck 1). ................................ ................... 30 6.3. Stress of strip of arch barrel due to live load (truck 2). ................................ ................... 30 6.4. Stress of strip of arch barrel due to l ive load (truck 3). ................................ ................... 31 6.5. Stress of strip of arch barrel due to live load (truck 4). ................................ ................... 31 6.6. Stress of strip of arch barrel due to load combination with truck at position 1. .............. 32 6.7. Stress of strip of arch barrel due to load combination with truck at position 2. .............. 32 6.8. Stress of strip of arch barrel due to load combination with truck at position 3. .............. 33 6.9. Stress of strip of arch barrel due to load combination with truck at position 4. .............. 33 6.10. Stress of strip of arch barrel due to dead and live load s. ................................ ............... 34 6.11. Arch dimensions (Robinson, 2000). ................................ ................................ .............. 34 6.12. The Ely Stone Bridge dimensions. ................................ ................................ ................. 35 6.13. Nomogram for determining the provisi onal axle loading of masonry arch bridges before factoring (The highways agency, 2001). ................................ ................................ ................. 37 6.14. Span/Rise factor (The highways agency, 2001). ................................ ........................... 38 6.15. Profile factor (The highways agency, 2001). ................................ ................................ . 39 6.16. The First position of the truck load in Ring. ................................ ................................ .. 44 6.17. The last position of the truck load in Ring. ................................ ................................ .... 44 6.18. Model of the Ely Stone Bridge in Ring. ................................ ................................ ........ 45 6.19. Model of the Ely Stone Bridge in Ring (3D view). ................................ ....................... 45 6.20. Moment diagram for one span assuming continuity of all stone unites. ....................... 46

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xi 6.21. Normal axial force diagram assuming continuity of all stone unites. ............................ 46 6.22. Joint coordinates for Risa 2D Frame Analsis. ................................ ............................... 47 6.23. Self weight load on the arch. ................................ ................................ ......................... 49 6.24. Fill load on the arch. ................................ ................................ ................................ ...... 50 6.25. Pressure load on the right side of the arch. ................................ ................................ .... 50 6.26. Pressure load on the left side of the arch. ................................ ................................ ...... 51 6.27. Axial force (kips) from dead load. ................................ ................................ ................. 51 6.28. Moment (kips ft) from dead load. ................................ ................................ .................. 52 6.29. The location of truck 1. ................................ ................................ ................................ .. 54 6.30. Axial force (kips) due to load combination of dead load and live load for truck in position 1. ................................ ................................ ................................ ................................ 54 6.31. Moment (Kips ft) due to load combination of daed load and live load for truck in position 1. ................................ ................................ ................................ ................................ 55 6.32. The location of truck 2. ................................ ................................ ................................ .. 57 6.33. Axial force (kips) due to to load combination of daed load and live load for truck in position 2. ................................ ................................ ................................ ................................ 57 6.34. Moment (kip s ft) due to to load combination of daed load and live load for truck in position 2. ................................ ................................ ................................ ................................ 58 6.35. The location of truck 3. ................................ ................................ ................................ .. 60 6.36. Axial force (kips) due to to load combination of daed load and live load for truck in position 3. ................................ ................................ ................................ ................................ 60 6.37. Moment (kips ft) due to to load combination of daed load and live load for truck in position 3. ................................ ................................ ................................ ................................ 61 6.38. The location of truck 4. ................................ ................................ ................................ .. 63

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xii 6.39. Axial force (kips) due to to load combination of daed load and live load for truck in position 4. ................................ ................................ ................................ ................................ 63 6.40. Moment (kips ft) due to to load combination of daed load and live load for truck in position 4. ................................ ................................ ................................ ................................ 64 6.41. Model of the arch with hinges in Risa 2D Frame. ................................ ......................... 65 6.42. Axial force (kips) due to load combination of dead load and live load for truck in position 1 with three hinges. ................................ ................................ ................................ ... 67 6.43. Moment (kips ft) due to load combination of dead load and live load for truck in position 1 with three hinges ................................ ................................ ................................ .... 67 6.44. Axial force (kips) due to load combination of dead load and live load for truck in position 2 with three hinges. ................................ ................................ ................................ ... 69 6.45. Moment (kips ft) due to load combination of dead load and live load for truck in position 2 with three hinges. ................................ ................................ ................................ ... 70 6.46. Axial force (kips) due to load combination of dead load and live load for truck in position 3 with three hinges. ................................ ................................ ................................ ... 72 6.47. Moment (kips ft) due to load combination of dead load and live load for truck in position 3 with three hinges. ................................ ................................ ................................ ... 72 6.48. Axial force (kips) due to load combination of dead load and live load for truck in position 4 with three hinges. ................................ ................................ ................................ ... 74 6.49. Moment(kips ft) due to load combination of dead load and live load for truck in position 4 with three hinges. ................................ ................................ ................................ ... 75 6.50. Finit element models. ................................ ................................ ................................ ..... 75 6.51. Self weight load Fill load in Risa FE. ................................ ................................ ............ 77 6.52. Pressure load on the right side of the arch in Risa FE. ................................ .................. 77 6.53. Pressure load on the left side of the arch in Risa FE. ................................ .................... 78

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xiii 6.54. Truck 1 position in Risa FE. ................................ ................................ .......................... 78 6.55. Truck 2 position in Risa FE. ................................ ................................ .......................... 79 6.56. Truck 3 position in Risa FE. ................................ ................................ .......................... 79 6.57. Truck 4 position in Risa FE. ................................ ................................ .......................... 80 6.58. The stress by dead load in Risa FE. ................................ ................................ ............... 80 ................................ .......... 81 ................................ .......... 81 ................................ .......... 82 ................................ ......... 82 7.1. Model of the Ely Stone Bridge in Risa 2D Frame. ................................ .......................... 84 7.2. Model of the Ely Stone Bridge i n Ring. ................................ ................................ .......... 85 .................. 85 7.4. Moment diagram by the critical design load in Risa 2D. ................................ ................ 86 7.5. Failure mechanism for the The Ely Stone Bridge from RING2.0. ................................ 86 7.6. Axial force diagram b y the critical design load in Ring. ................................ ................. 87 7.7. Moment diagram by the critical design load in Ring. ................................ ...................... 87 7.8. Model of the Ely Stone Bridge under critical load in Risa 2D Frame with three hinges. 88 7.9. Axial force (kips) diagram by the critical design load in Risa 2D with three hinges. ..... 88 7.10. Moment (kips ft) diagram by the critical design load in Risa 2D with three hinges. .... 89

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1 CHAPTER I 1 INTRODUCTION 1.1 Introduction Stone masonry arch bridges are one of the oldest forms of structures. Many have performed for centuries. Many historic stone masonry arches now carry live loads much greater than their original intended purpose ( Lily Beyer, 2012 ). The U.S. has far fewer stone arch bridges than European countries . Many of these stone arch bridges were built pr ior to 1910; one of these is the Ely Stone B ridge in Jones County, Iowa . Its builders could not have predicted the modern day changes of width, truck load, guard rails, drai nage, erosion, and the long term effects of flooding and freez e/thaw cycles . 1.2 Research Objective The main goal of this thesis is to study and assess the Ely Stone Arch Bridge in light of current traffic loads , and to apply multiple analytical methods to t he Ely Stone Bridge. The bridge will be modeled analytically using the following methodologies: The MEXE Method Ring 3.2 Risa 2D Frame Analysis Risa 2D Finite Element Method (FEM) 1.3 Outline o f T his T hesis T he contents of each chapter are briefly described below: Chapter I : T his chapte r will show a n overv iew of the masonry arch bridge, statement of the research objective, and outline of this thesis.

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2 Chapter II : T his chapte r will present a brief a literature review . Chapter III : T his chapte r will have d escription of the Ely Stone B ridge. Chapter IV : This chapter will discuss different types of assessment software such as MEXE method, Ring method , Risa 2D Frame method, and Risa 2D Finite Element method . Chapter V : This chapter will p resent the loads that are applied on the arch and parameters of the bridge. Chapter VI : This chapter will i llustrate the analysis of the bridge resulting from different methods . Chapter VII : This chapter will provide a comparison of different calculation methods, conclusion, and recommendations for future work .

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3 CHAPTER II 2 LITERATURE REVIEW 2.1 Introduction There are approximately 1700 masonry stone arch bridges in the United States according to 2013 National Bridge Inventory database (NBI, 2013). Approximately , 50% of these bridges were constructed before 1910 and approximately 50 % of them are still in service more than 100 years later. On the other hand, the same study shows that only 4% of the steel bridges, and 1% of the concrete bridges built before 1910 remain in service. Since most early settlers in the United States were of European heritage, they had the skills required to build arch bridges (Citto & Woodham, 2016). 2.2 History of Stone Masonry Arch Bridge Bridges are an important part of the transpor tation infrastructure . M any centuries ago, the masonry stone arch bridge was a common form of construction . The original form of stone masonry bridges goes back to a ncient time s , most notably the Romans (Boyd, 1978). The Greeks were the first to use the arch in structures. There is an example of a Greek arch bridge which is a very small bridge in Rhodes. On the other hand, the Romans built great arch bridges and used them as aqueducts to trans fer water to their cities (Boyd, 1978). In the M iddle A ges, after the fall of the Roman Empire, building bridges became more important than ever before. The reason for this development is related to armies needing bridges to move troops and supplies. However, with this increased construction of bridges, the en gineering knowledge, and economic resources required to build these bridges became more commonplace (Boyd, 1978).

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4 At the present time, there are many bridges, built in the ni neteenth century and earlier , that are still functioning and still serving critica l roles in roadway and railway networks around the world . However, these bridges are likely carrying traffic loads much higher than the load for which they were originally intended (Boyd, 1978). Even though the structural behavior of masonry arch bridg e a ppears simple and easy at first, their modelling and mechanical behavior is complicated because of nonlinear effects. In addition, even if no new masonry arch bridges are built , the restoration of existing bridges is complicated by the use of both old an d new materials and te chniques (Boyd, 1978). 2.3 A rches The arch design is one of the oldest forms of bridges and has great strength. Before 20th century, the arches were used to span the distance between two su pports such as walls or piers a nd they were constru cted by using stone and bricks. In the 20th century, arches are commonly used in bridge construction a nd are constructed us ing steel and reinforced concrete . Arches are resistant to loads that inclu de live load and self weight of the bridge to the abutments at each end, where those ends resist the horizontal thrust. So, the abutments of the arch should be large to resist these forces. However, masonry arches need to be designed for stability because th ey are subjected to separa tion of joints in regions of high flexural tension.

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5 Figure 2 . 1 . Distribution forces in the arch . A curved beam faces horizontal and vertical reaction s . So, the support at the two ends must not be a roller. These types of arches do not develop horizontal ly at the base like a corbelled arch. A corbelled arch was common in ancient civiliza tions and in the Americas. T his kind of arch can develo p bending stress in its members (Drysdale and Hamid, 2008) . Figure 2 . 2 . Corbelled a rch. Corbelling masonry dates back to 2900 B.C. An example of a corbel arch can be found in the Mycena ean fortress at Tiryns , Greece . Corbelling uses subsequent courses of stone or brick on each side of the openi ng, where each course protrude s beyond the previous

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6 course. These courses of protruding materials were supported until the entire arch was completed (Drysdale and Hamid, 2008) . The arch shape i s important for stone masonry bridges because this form can easily develop compressive stress. C ompressive stress is a positive feature because the stones in the arch are effectively prestressed in compression. For this reason, it was a preferred shape fo r a long time (Beyer, 2012). T he shape of the arch was made of stones which were cut into a trapezoidal form. T he keystone is important component in creating the arch shape . In addition, the arch can be made from bricks bonded with mortar. T his type of arch is common in many countries especially in the United Kingdom . 2.3.1 Arches classified. There are three types of arches which are usually used: Fixed Fixed Arch Two Hinged Arch Three Hinged Arch Fixed fixed arch es and two hinged arch es are stat ically indeterminate structures. However, the reaction s and internal forces can be calculated by flexibility matrix method. T he three hinged arch is a statically determinate structure which means that the reactions and all internal forces can be calculated by static equations of equilibrium ( Three Hinged Arch ).

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7 Figure 2 . 3 . Fixed f ixed a rch . Figure 2 . 4 . Two h inged a rch . Figure 2 . 5 . Three h inged a rch .

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8 2.3.2 B ehavior of the a rch under load . When dead load is applied , the arch will be very stable. especially if the centerline of the arch is very close to the line of thrust. I n large arches, the live load is smaller than the dead load. So, the effects of live load will be limited. On the other hand, live load will be more predominate than dead load on small arch structures. W hen an unbalanced live load is applied on the arch, it will cause a bending moment in the rib , or barrel of the arch. I n this situation, the arch must be designed to resist the additional load so the arch will remain stable u nder the unbalanced load (Lily Beyer, 2012). 2.4 Description of the Stone Masonry Arch Bridge Masonry arch bridges typically have different components that include arch, abutment, backfill, and the wing walls . It also has one or more rings of stone as seen in F igure 2. 6 . Figure 2 . 6 . Important parts of an arch bridge. As shown in Figure 2. 7 , a masonry arch bridge has two basic component elements which are the arch and parapet.

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9 Figure 2 . 7 . Arch and parapet in bridge. Stone is the main component in many masonry arch bridge s but it is very weak in tension. Also, joints between stone units tend to open when in tension. So, the material used in masonry arch bridge s , has very low tensile strength and has great ability to resist compression. T he strength in masonry arches is dependent on the strength and stability of arch barrel. 2.4.1 M aterials. Stone b ridge s are often made from sandstone or limestone. S andstone is sedimentary debris rock and it has resistance to compression between 3.5 to 14.5 ksi . Limestone is carbonated rock and it has resistance to co mpression between 3 to 11.5 ksi ( Martinez et al, 200 1). 2.5 S tructural Analysis of Old Masonry Arch Bridge T he structural analysis of old masonry bridge s is not an easy or simple matter. T hese bridges were intended for live load s common in their construction time. However, they are now carrying loads beyond those envisioned by their designer.

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10 T here are many reasons that make modeling this type of bridge difficult , e.g. the behavior of masonry arches is not commonly found in some software. Also, the information and history of this bridge may not be complete .

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11 CHAPTER III 3 THE ELY STONE BRIDGE 3.1 Introduction Stone masonry arch bridges are one of the oldest forms of structures. This kind of bridge has many issues and questions related to the modeling and analysis . In addition , most of these bridges increasingly deteriorate. For instance, t he Ely Stone Bridge in J ones C ounty, Iowa, is a 100 year old stone masonry bridge located in the United S tates. It has successfully served for many years. This bridge has several condition issues; however, it is still in service, and it remains as a part of the history of the county . It is only natural for a community to want to preserve the beautiful and fun ctional Ely Stone Bridge given its history and heritage in the town of Monticello, Iowa. 3.2 Background and Description In 1893, Reuben Ely Sr . and his son Reuben Ely Jr . constructed the Ely Stone Bridge , which has three elliptical stone arches, near Monticello in Jones County , Iowa . They used native stone which was brought in from different pl aces in Iowa. The limestone used for the voussoir was brought in from Anamosa, Iowa, and the stone that is located on the side spandrels and piers was brought in from streambeds . The stone from Anamosa, Iowa , which was used to build the arches of the bridge , was located about twelve miles from the construction site. The Ely Stone Bridge measures approximately 68 feet long, 15.5 feet wide and crosses over the Wet C reek. Also, it has three spans of masonry elliptical arches. Each span measures approximately 20 feet in length and each arch span has double layers of limestone units called voussoirs . There are vertical piers which are approximately 5 feet high.

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12 Like a modern bridge, the top of the Ely Stone Bridge has pavement which is a reinforced concrete slab. It is approximately 8 inches thick and around 16 feet wide. In addition, the Ely Stone Bridge has been listed in the National Register of His toric Places in 1979 ( NRHP 1979, Jackson 1988) . Figure 3 . 1 A photograph of the bridge when it was registered as a historic place.

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13 Figure 3 . 2 . Satellite view of the Ely Stone Bridge. Photographs of the Ely Stone Bridge are shown in Figures 3.3 and 3.4 below: Figure 3 . 3 . Side view of the Ely Stone B ridge at east side.

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14 Figure 3 . 4 . Side view of the Ely Stone B ridge at west side. ( For more detailed information about the Ely Stone Bridge , see Appendix A )

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15 CHAPTER IV 4 ANALYSIS METHODS : GENERAL 4.1 Introduction T here are numerous ways to model and analyze a stone arch bridge. These methods have been develop ed since antiquity, through the inception of modern structure analysis in the 19 th century, and continue to this day . In the past, the method s for structural assessment were ba sed on approximate calculation method s . T hey only provide an approximate estimate of the capacity load. After years, tremendous progress provided more advanced numerical modelling strategies, and t he analysis, description, and geometry of the bridge was more accurate. Advanc es in technology developed new software which can create models using finite element modeling methods. There are two and three dimensional finite ele ment models. In this chapter, various methods are presented and compared their results. However, these met h ods are not a complete inventory . 4.2 Methods U sed for Modeling There are two different levels of analysis used in this paper to examine the condition of the Ely Stone B ridge. The first level is defined as the approximate calculations method and it still widely used . This level of analysis contains methods such as : T he g raphical analysis method T he t hree hinged arch method T he MEXE method. At the second level, there is a simple 2 D modelling such as : Th e Risa 2D Frame Analysis method The Risa 2D Finite Element method

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16 The Ring me thod 4.2.1 Three Hinged Arch m ethod From the name of this method, it is clear to understand that Three Hinged Arch method based on three hinge s . There are two hinge s at the supports, which are located on the end of the arch, and the thi rd hinge will be the support at the crown of the arch. This method is based on the calculation of the bending moment in a three pinned arch. It is a simple and easy method because the reactions on the arch can be determine d by using the forces on each direction equal to z ero. A three pinned based on simple statics method . T he stress in the member can be calculated by determined the bending moment in this member ( Three Hinged Arch ). In order t o calculate any internal moment a t any point on an arch, the r eaction on t he abutments must first be calculated. In this type of arch, there are f our reactions a nd all of them are unknown. However, there are three static equations equilibrium and one moment equation . The moment equation, determine the moment a round the third hinge for all forces on the arch , on either side of it , to calculate the r eactions ( Three Hinged Arch ). Figure 4 . 1 . Three hinged arch .

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17 4.2.2 The M odified MEXE method The M odified MEXE m ethod is commonly used, particularly in the UK. This method was developed by the Military Engineering Experimental Establishment, MEXE . The MEXE method is as a considered approximate calculation method. However, it is f ast and easy in comparison with other methods (Lasell and Bjurstrom, 2009). There are some limitations in order to be abl e to use this method : T he arch should not be appreciably deformed or have a flat shape . T he span of brid ge must not be longer than 18m (59ft). T he skew of the bridge must be smaller than 15 degrees . T he backfill above the extrados has to be less than 1m (3.3ft). T he bridge does not have a multi span (except if the ratio of the arches work separately and the ratio of the height to width piers of multi span is smaller than 2). The target of the MEXE method is to calculate the provisional axle load (PAL) capacity of masonry arches . So, PAL based on the span, the average depth between the arch at the crown and the surface of road, and th e th ickness of the arch barrel ( Robinson BEng , 2000). The following Definition of arch bridge dimensions as shown in Figure 4.2. r c The rise of the arch at the crown (m) r q The rise of the arch at the quarter points (m) d T he thickness of the arch barrel to the keystone (m) h The depth of the fill (m)

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18 Figure 4 . 2 . Arch bridge dimensions for MEXE method ( Robinson BEng , 2000) . 4.2.3 RING 3.2 Ring 3.2 is software which can assess single and multiple span masonry arch bridge s . It can calculate the maximum safe load that ca n be applied on the arch bridge . Also, it is able to determine the critical failure load that impact s the arch bridge. Ring considers the bond between the blocks with the help of friction coefficients. The analysis method used in Ring can work with bridge s that face little damage, bridge s that have lost their mortar , and masonry material. T he program shows the thrust line with failure model at the same time ( LimitState, 2009) .

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19 Figure 4 . 3 . S hows explanations a RING2.0 analysis ( l imitState, 2009) . 4.2.4 RISA Risa is a suite of program s that are used widely in structural analysis and design in the United States. Risa software can be used for analysis of tu nnels, roller coasters, soccer stadiums, building s, etc . In this study, Risa 2D Frame analysis and Risa 2D F inite Element analysis are used (RIS A, 2013). 4.2.5 Three dimensional FEM A masonry arch bridge can be modeled in three dimensions. This analysis can model the entire structure which included abutments, spandrel walls. We might expect this kind of analysis to provide more realistic results and giv e better load carrying capacity . However, three dimensional finite element (FEM) analysis is difficult to use for masonry unites when joint separations occur.

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20 CHAPTER V 5 LOADS 5.1 Introduction Loads largely affect the analysis of a structure. Applying different dead load s over a structure and different locations of live load s can largely e ffect the d esign. In this study, dead load, live load , and earth pressure load have been considered. Neither snow load nor temperature load were considered . 5.2 Design Cod es and Regulations In the United States, AASHTO LRFD 2012 Bridge Design Specifications is typically used for bridge design . 5.3 Loads Dead load, live load, and earth pressure load hav e been applied to analyze of the Ely Stone Bridge. For this study, l ive load was applied at four positions on the bridge . Based on that, five load combinations were used : Dead load only Dead plus truck location 1 Dead plus truck location 2 Dead plus truck location 3 Dead plus truck location 4 A ll loads will be applied on t he strip of arch barrel as shown in F igure 5.1 below:

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21 Figure 5 . 1 . Illustration of arch barrel strip. 5.3.1 Dead load . D ead load include s the weight of structure, fill load, and lateral pressure load. variable depth, and a coefficient of active earth pressure of 0.3. In fact, the arch weight lead s to the creation of stress into its members. T he self weight of the arch has been included in all analysis . The stresses that are created from dead load is mostly compression stress with some bending stress in the arch. T o calculate fill load along the arch, the depth for each point along the x axis of the arch multiplied by f ill's s elf weight ( ): 0.120 k ips per cubic foot ( k cf ) . B ecause the depth of the arch changes over t he lengt h of the bridge, this causes change in the value of fill load as seen in Fi gure 5.2 . PARAPET INITIAL ANALYSES ON 2D STRIP OF ARCH BARREL ARCH BARREL

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22 Figure 5 . 2 . Weight of fill along the strip of arch barrel . 5.3.2 L ive l oad t ruck l Live load is a variable load, and it induces the maximum stress when it is located over only part of the span. In this paper, live load was applied as a distributed load by the area on the strip of the arch barrel . However, it will be covered specific area on the strip depended the depth of the fill . The truck load was applied in four positions as shown in Figures below: Figure 5 . 3 . . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 Fill Load (k/ft2) length of the arch (ft)

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23 Figure 5 . 4 . location . Figure 5 . 5 . 3

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24 Figure 5 . 6 . 5.4 Load Combinations Combining all loads, which include self weight of the arch, fill load, pressure load, and live load, will create stress es as shown in Figures below: 5.4.1 Combined dead load and live loads The load combination which includes s elf w eight, f ill load, live load ( Truck load ), and p ressure load. The distribution of all loads on all the arch shown in Figure s below:

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25 Figure 5 . 7 . Load combination with truck at position 1. This load combination will have s elf w eight, f ill load, live load (t ruck position 1 ), and p ressure load as seen in Figure 5.7.

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26 This load combination will have s elf w eight, f ill load, live load (t ruck position 2 ), and p ressure load as seen in Figure 5.8. Figure 5 . 8 . Load combination with truck at position 2.

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27 Figure 5 . 9 . Load combination with truck at position 3. This load combination will have s elf w eight, f ill load, live load (t ruck position 3), and pressure load.

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28 Figure 5 . 10 . Load combination with truck at position 4. This load combination will have s elf w eight, f ill load, live load (t ruck position 4), and pressure load.

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29 CHAPTER VI 6 ANALYSIS AND COMPARSION OF THE RESULTS The goal of this paper is to compare different methods for analyzing masonry stone arch bridges. Th e comparisons have been made with different methods to calculate the ultimate load capacity using the same arch, under the same loads . 6.1 Linearly Elastic Analysis From the loads describe d in chapter 5, an d assuming linear elasticity applies (i.e. no hinges). Then stresses in the barrel of the arch can be determined . To calculate the stres s in each section of the arch due to dead load we need to know s hape s ection, axial force, and the moments that are cre ated by dead load for each of the node on the arch. T he summation of the axial force is divided by the area of the stone and the moment d ivided by t he section modulus . From F igure 6.1 , the max imum stress occurs close to the ends, where the arch cross section area is under the maximum dead load . Figure 6 . 1 . Stress of strip of arch barrel due to dead load (self wight, fill). In addition, the stress in each segment on the arch due to live load at each position, the live load is distributed as a concentrate d load on the top of the arch to a distributed load -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 25 Stress (ksi) length of the arch (ft)

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30 on the strip of the arch barrel . The moment and axial force along the strip of the arch are calculated to determine the stresses on all sections. The f ollowing F igures illustrate the stresses due to truck load on the strip of the arch at four positions . Figure 6 . 2 . Stress of strip of arch barrel due to live load (truck 1). Figure 6 . 3 . Stress of strip of arch barrel due to live load (truck 2). -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0 5 10 15 20 25 Stress (ksi) length of the arch (ft) -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0 5 10 15 20 25 Stress (ksi) length of the arch (ft )

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31 Figure 6 . 4 . Stress of strip of arch barrel due to live load (truck 3). Figure 6 . 5 . Stress of strip of arch barrel due to live load (truck 4). -0.030 -0.020 -0.010 0.000 0.010 0.020 0.030 0.040 0.050 0 5 10 15 20 25 Stress (ksi) length of the arch (ft) -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0 5 10 15 20 25 Stress (ksi) length of the arch (ft)

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32 The stresses of all loads combination shown in Figures below: Figure 6 . 6 . Stress of strip of arch barrel due to load combination with truck at position 1. Figure 6 . 7 . Stress of strip of arch b arrel due to load combination with truck at position 2. -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 0.100 0 5 10 15 20 25 Stress (ksi) length of the arch (ft) -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 0.100 0 5 10 15 20 25 Stress (ksi) length of the arch (ft)

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33 Figure 6 . 8 . Stress of strip of arch barrel due to load combination with truck at position 3. Figure 6 . 9 . Stress of strip of arch barrel due to load combination with truck at position 4. -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 0.100 0 5 10 15 20 25 Stress (ksi) length of the arch (ft) -0.020 0.000 0.020 0.040 0.060 0.080 0.100 0 5 10 15 20 25 Stress (ksi) length of the arch (ft)

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34 Figure 6 . 10 . Stress of strip of arch barrel due to dead and live load s . 6.2 Analysis U sing MEXE The modified MEXE method is used to determine the carrying capacity of arches. The f irst step in this method is to calculate PAL, then it must be factored depend ing on material factor, joint factor, p rofile factor, and span factor . Dimensions used in the MEXE method analysis are shown in Figure 6. 11 . Figure 6 . 11 . Arch dimensions ( Robinson , 2000). -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 0.100 0 5 10 15 20 25 Stress (ksi) length of the arch (ft) Load combination with truck 1 Load combination with truck 2 Load combination with truck 3 Load combination with truck 4

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35 L The length of the span (m) r c .. Ratios to intrados of semi circular arch (m) r q ... R i se at L/4 (m) d (m) h (m) This result of the MEXE method for the Ely Stone Bridge based on the dimensions that are shown in the F igure 6. 1 2 below : Figure 6 . 1 2 . The Ely Stone Bridge d imensions . Where L= 19.82ft= 6.0 m r c = 5.6 ft = 1.7 m rq = 3 ft = 0.9 m h = 2.15 ft = 0. 7 m d= 2ft= 0.6 m

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36 6.2.1 Calculate t he provisional axle load. The provisional axle loading can be obtained by following equation : ton n e s) Note : 1 (m etric) tonne = 1.023 (U.S) ton. The provisional axle load can be determined by the nomogram by simply drawing three straight lines (A, B, and C) as shown in Figure 6.3 below. Mark the (L ), which is arch span, on column A, and mark ( d + h ), which is the total crown thickness (barrel and fill), on column B then draw line to column C. The number that faces the line will be the PAL in tons . Even by using the equation or the nomogram , the result has to be the same (The highways agency, 2001). By using the e quation : = 114.3 Use PAL= 70.0 ton ne s T his value of the provisional axle load must be modified by multiple factors . By using the following nomogram L=6.0 m d+h= 1.3 m Line through these points to column C PAL= 70.0 ton ne s

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37 Figure 6 . 13 . Nomogram for d etermining the p rovisional a xle l oading of m asonry a rch b ridges before f actoring (The highways agency, 2001).

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38 6.2.1.1 Calculate m odification f actors. 1. Span /Rise Factor ( ): this represents the arch shape. When a span/rise ratio is 4 or less, then it will be assumed to give optimum strength and the factor assumed to be 1 . However, if the ratio between span and rise is greater than 4, Figure 6. 1 4 will be u sed to give the appropriate span/ rise factor ( Fsr ) . Where the equation of span/rise factor cal culated by following expression (The highways agency, 2001). Figure 6 . 14 . Span/Rise f actor (The highways agency, 2001). = 3.5 Then take = 1.0 2. Profile Factor ( ): this factor is based on the fact that segmental and parabolic arches are stronger than elliptical ar ches of similar span/rise ratio and thickness of barrel . T he profile factor ( ) ca n be calculated even by Fig 6 . 1 5 or by the following equation (The highways agency, 2001).

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39 F = 1. 5 Thake F = 1 Also, from F igure below, F = 1.0 Figure 6 . 15 . Profile f actor (The highways agency, 2001). 3. Material Factor ( Fm ): This factor focuses on the depth of the fill and the height of the barrel. The material factor calculated by equation below (The highways agency, 2001). Where the values for F and Ff can be determine from T ables 6 .1 and 6 .2 respectively . 1.0 0.5 0 0.70 0.75 0.80 0.85 0.90 0.95 1.0 r q r c

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40 Table 6 . 1 . Barrel factor (The highways agency, 2001). Arch Barrel Barrel Factor ( Fb ) Granite whether random or coursed and all built in course masonry except limestone, all with large shapes voussoirs 1.5 Ashlar quality siliceous sandstone 1.4 Concrete # or engineering bricks and similar sized masonry (not limestone). 1.2 Limestone, whether random or coursed, ashlar quality calcareous sandstone, good random masonry and building bricks, all in good condition. 1.0 Masonry of any kind in poor condition (many voussoirs flaking or badly spalling, shearing etc.). Some discreti on is permitted if the dilapidation is only moderate. 0.7 From T able 6 .1, a ppropriate values of the barrel factor F b = 1.0 Table 6 . 2 . Fill factor (The highways agency, 2001). Filling The Fill Factor ( F f ) Concrete # 1.0 Grouted materials (other than those with a clay content) 0.9 Well compacted materials* 0.7 Weak materials evidenced by tracking of the carriageway surface 0.5 And from T able 6 .2, a ppropriate values of the fill factor Ff = 0. 7 Apply these values in the last equation and determine Fm Fm = 1.0 4. Joint Factor ( Fi ) : this factor is based on the size and situation of joints and general size. It is calculated from the following formula (The highways agency, 2001).

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41 Where : Fw = is the w idth factor Fd = is the d epth factor Fmo = is the mortar factor. All these factors can be acquired from Tables 6.3 , 6 .4 , and 6 . 5 respectively. Table 6 . 3 . Width factor (The highways agency, 2001). Width of Joint Width Factor ( Fw ) Joints with widths up to 6mm 1.0 Joints with widths between 6mm and 12.5mm 0.9 Joints with widths over 12.5mm 0.8 From T able 6 .3, a ppropriate values of the width factor Fw = 0.8 Table 6 . 4 . Depth factor (The highways agency, 2001). Construction of Joint Depth Factor ( F d ) Unpointed joints, pointing in poor condition and 0.9# joints with up to 12.5mm from the edge insufficiently filled Joints with from 12.5mm to one tenth of the thickness of the barrel insufficiently filled 0.8# Joints insufficiently filled for more than one tenth At the + the thickness of the barrel discretion From T able 6 . 4 , a ppropriate values of the depth factor F d = 1.0

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42 Table 6 . 5 . Mortar factor (The highways agency, 2001). Condition of Joint Mortar Factor ( F mo ) Mortar in good condition 1.0 Loose or friable mortar 0.9 From T able 6 .5, a ppropriate values of the mortar factor F m o = 1.0 = = 0 .8 5. Condition Factor ( ): This factor is based on an assessment of the importance of the various cracks and deformations. The range of this factor is between 0 and 1.0 based on t he stability and load carry ing capacity of the arch barrel. Where equal 1.0 when the arch has good condition with no defects . However, equal 0 when the bridge has poor condition (The highways agency, 2001). = 1.0 6.2.2 Modified axle load . To determine the modified axle load that shows the allowable loading on the arch, we must apply all previous factors and multiply by the provisional axle loading as given in following expression (The highways agency, 2001). = 1.0 × 1 .0 × 1.0 × 0. 8 × 1.0 × 70 = 54 ton ne s = 120 kip s For more information about this method, see Appendix B .

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43 6.3 Analysis U sing Ring . RING 3.2 software analysis was performed for the Ely Stone Bridge. The Ely Stone Bridge was modelled with 18 ft ( 5791 mm) width, 19.82 ft (6000 mm) a span arch bridge, 2.15 ft ( 7 00 mm) a fill height over arch crown, and 5.6 ft ( 170 7 mm) th e rise of the arch at the crown . In addition, the bridge was analy zed with the parameters that are presented in Table 6.6 below : Table 6 . 6 . Parameters used in Ring for the Ely Stone Bridge . Material Properties Masonry Backfill Unit Weight (K N /m ³ ) 25 19 Friction coefficient 0.6 0.6 Angle of internal friction 30 A single axle loads was applied spread apart to make sure our results were positioned over the span with value of 1 KN and again with a value of 112.82 KN. T hey gave the same critical load at the same location . However, all load information is presented in T able 6.7. The critical design load happened with a single axle load as shown below: Table 6 . 7 . Load f actor used in Ring for the Ely Stone Bridge. Load kind Adequacy factor C ritical design load Position of the load from support Single Axle (1kN) 977 220 k ip 3900 mm EU Single Axle (112.82 KN ) 8.66 220 kip 3900 mm

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44 Figure 6 . 16 . The First position of the truck load in Ring. Figure 6 . 17 . The last position of the truck load in Ring.

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45 Figure 6 . 18 . Model of the Ely Stone Bridge in Ring. Figure 6 . 19 . Model of the Ely Stone Bridge in Ring ( 3D view ) .

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46 Figure 6 . 20 . M oment diagram for one span assuming contin uity of all stone unites. Figure 6 . 21 . Normal axial force diagram assuming continuity of all stone unites. To see more details of this method, Appendix C provides all analysis results for one span and three spans .

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47 6.4 A nalysis U sing Risa 2D Frame . Risa 2D Frame analysis is based on plane frame analysis. U sing the frame analysis means using the simplest form of modelling. In this analysis, the frame usually used fixed supports on the abutments. The Ely Stone Bridge was modelled with different joint coordinates on a strip of arch barrel. W here the strip was divided in to ma ny elements based on different dimension s in X a nd Y as shown in F igure below Figure 6 . 22 . Joint coordinates for Risa 2D Frame Analsis .

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48 Also, the bridge performed with the parameters illustrate d in the T able below: Table 6 . 8 . Parameters used in Risa 2D Frame for the Ely Stone Bridge. Material Properties Arch ( Limestone ) E Elastic Modulus ( ksi) 4400 G Shear Modulus (ksi) 1760 Poisson's Ratio 0.25 kcf) 0.160 Fill's Self Wight ( kcf ) 0.120 6.4.1 Loads The loads applied on the arch are dead load, which include self weight load, fill load, and pressure load, and live load, which include truck load on four different positions as shown in the T able and F igures below: Table 6 . 9 . Dead load values in Risa 2D Frame. N ode # Self Wight Load (k/ft) Fill Load (k/ft) Pressure Load (Right) (k/ft) Pressure Load (Left) (k/ft) N1 0.32 1.17 0.351 N2 0.32 0.92004 0.276012 N3 0.32 0.738 0.2214 N4 0.32 0.63 0.189 N5 0.32 0.5508 0.16524 N6 0.32 0.498 0.1494 N7 0.32 0.45 0.135 N8 0.32 0.4248 0.12744 N9 0.32 0.405 0.1215 N10 0.32 0.39 0.117 N11 0.32 0.38004 0.114012 N12 0.32 0.375 0.1125 N13 0.32 0.38004 0.114012

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49 N14 0.32 0.39 0.117 N15 0.32 0.405 0.1215 N16 0.32 0.4248 0.12744 N17 0.32 0.45 0.135 N18 0.32 0.498 0.1494 N19 0.32 0.5508 0.16524 N20 0.32 0.63 0.189 N21 0.32 0.738 0.2214 N22 0.32 0.92004 0.276012 N23 0.32 1.17 0.351 Figure 6 . 23 . Self weight load on the arch .

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50 Figure 6 . 24 . Fill load on the arch . Figure 6 . 25 . Pressure load on the right side of the arch .

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51 Figure 6 . 26 . Pressure load on the left side of the arch . The Figures below show the results which are axial force and moment from dead load: Figure 6 . 27 . Axial force (kips) from dead load.

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52 . Figure 6 . 28 . Moment (kips ft ) from dead load. T he RIS A 2D frame analysis uses four different load c ombinations based on four positions of the live load. T he description and result for each load combination is shown on the Tables below: 6.4.1.1 Load combination with truck in position 1 This load combination is self weight load, fill load, p ressure load, and l i ve load for truck 1. Table 6 . 10 . Total stress due to load combination with truck in position 1. Member Label Section Axial[ k ] Moment [ k ft] Axial ( ksi ) Top Bending ( ksi ) Bot Bending ( ksi ) Total Stress ( ksi ) Total Stress ( psi ) M1 I End 13.0 3.17 0.05 0.03 0.03 0.01 12 j End 12.3 2.09 0.04 0.02 0.02 0.06 65 M2 I End 11.9 2.09 0.04 0.02 0.02 0.06 63 j End 11.4 5.08 0.04 0.05 0.05 0.09 93 M3 I End 11.2 5.08 0.04 0.05 0.05 0.09 92 j End 10.9 4.71 0.04 0.05 0.05 0.09 87

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53 M4 I End 10.6 4.71 0.04 0.05 0.05 0.09 86 j End 10.4 3.57 0.04 0.04 0.04 0.07 73 M5 I End 10.0 3.57 0.03 0.04 0.04 0.07 72 j End 9.9 1.62 0.03 0.02 0.02 0.05 51 M6 I End 9.8 1.62 0.03 0.02 0.02 0.05 51 j End 9.6 0.23 0.03 0.00 0.00 0.04 36 M7 I End 9.3 0.23 0.03 0.00 0.00 0.03 35 j End 9.3 1.96 0.03 0.02 0.02 0.01 12 M8 I End 9.2 1.96 0.03 0.02 0.02 0.01 12 j End 9.2 3.76 0.03 0.04 0.04 0.01 7 M9 I End 9.2 3.76 0.03 0.04 0.04 0.01 7 j End 9.1 5.18 0.03 0.05 0.05 0.02 22 M10 I End 9.2 5.18 0.03 0.05 0.05 0.02 22 j End 9.1 6.06 0.03 0.06 0.06 0.03 31 M11 I End 9.2 6.06 0.03 0.06 0.06 0.03 31 j End 9.1 6.21 0.03 0.06 0.06 0.03 33 M12 I End 9.1 6.21 0.03 0.06 0.06 0.03 33 j End 9.2 6.06 0.03 0.06 0.06 0.03 31 M13 I End 9.1 6.06 0.03 0.06 0.06 0.03 31 j End 9.2 5.18 0.03 0.05 0.05 0.02 22 M14 I End 9.1 5.18 0.03 0.05 0.05 0.02 22 j End 9.2 3.76 0.03 0.04 0.04 0.01 7 M15 I End 9.2 3.76 0.03 0.04 0.04 0.01 7 j End 9.2 1.96 0.03 0.02 0.02 0.01 12 M16 I End 9.3 1.96 0.03 0.02 0.02 0.01 12 j End 9.3 0.23 0.03 0.00 0.00 0.03 35 M17 I End 9.6 0.23 0.03 0.00 0.00 0.04 36 j End 9.8 1.62 0.03 0.02 0.02 0.05 51 M18 I End 9.9 1.62 0.03 0.02 0.02 0.05 51 j End 10.0 3.57 0.03 0.04 0.04 0.07 72 M19 I End 10.4 3.57 0.04 0.04 0.04 0.07 73 j End 10.6 4.71 0.04 0.05 0.05 0.09 86 M20 I End 10.9 4.71 0.04 0.05 0.05 0.09 87 j End 11.2 5.08 0.04 0.05 0.05 0.09 92 M21 I End 11.4 5.08 0.04 0.05 0.05 0.09 93 j End 11.9 2.09 0.04 0.02 0.02 0.06 63 M22 I End 12.3 2.09 0.04 0.02 0.02 0.06 65 j End 13.0 3.17 0.05 0.03 0.03 0.01 12

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54 Figure 6 . 29 . The location of truck 1 . Figure 6 . 30 . Axial force (kips) due to load combination of dead load and live load for truck in position 1 .

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55 Figure 6 . 31 . Moment (Kips ft) due to load combination of daed load and live load for truck in position 1. 6.4.1.2 Load combination with truck in position 2. Description: ( Self Weight + Fill+ Truck 2 + Pressure 1+Pressure2 ) Table 6 . 11 . Total stress due to load combination with truck in position 2 Member Label Section Axial[k] Moment [k ft] Axial (ksi) Top Bending (ksi) Bot Bending (ksi) Total Stress(ksi) Total Stress (psi) M1 I End 13.2 2.15 0.05 0.02 0.02 0.02 23 j End 12.5 2.58 0.04 0.03 0.03 0.07 70 M2 I End 12.1 2.58 0.04 0.03 0.03 0.07 69 j End 11.6 5.11 0.04 0.05 0.05 0.09 93 M3 I End 11.3 5.11 0.04 0.05 0.05 0.09 92 j End 11.0 4.35 0.04 0.05 0.05 0.08 83 M4 I End 10.6 4.35 0.04 0.05 0.05 0.08 82 j End 10.4 2.86 0.04 0.03 0.03 0.07 66 M5 I End 10.0 2.86 0.03 0.03 0.03 0.06 64 j End 9.9 0.57 0.03 0.01 0.01 0.04 40 M6 I End 9.7 0.57 0.03 0.01 0.01 0.04 40 j End 9.6 1.09 0.03 0.01 0.01 0.02 22 M7 I End 9.2 1.09 0.03 0.01 0.01 0.02 21 j End 9.1 3.27 0.03 0.03 0.03 0.00 2

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56 M8 I End 9.1 3.27 0.03 0.03 0.03 0.00 3 j End 9.0 4.76 0.03 0.05 0.05 0.02 18 M9 I End 9.0 4.76 0.03 0.05 0.05 0.02 18 j End 8.9 5.54 0.03 0.06 0.06 0.03 27 M10 I End 9.0 5.54 0.03 0.06 0.06 0.03 27 j End 8.9 5.67 0.03 0.06 0.06 0.03 28 M11 I End 9.0 5.67 0.03 0.06 0.06 0.03 28 j End 9.0 5.19 0.03 0.05 0.05 0.02 23 M12 I End 9.0 5.19 0.03 0.05 0.05 0.02 23 j End 9.1 4.73 0.03 0.05 0.05 0.02 18 M13 I End 9.0 4.73 0.03 0.05 0.05 0.02 18 j End 9.0 3.93 0.03 0.04 0.04 0.01 10 M14 I End 9.0 3.93 0.03 0.04 0.04 0.01 10 j End 9.0 2.78 0.03 0.03 0.03 0.00 2 M15 I End 9.0 2.78 0.03 0.03 0.03 0.00 2 j End 9.1 1.23 0.03 0.01 0.01 0.02 19 M16 I End 9.1 1.23 0.03 0.01 0.01 0.02 19 j End 9.2 0.71 0.03 0.01 0.01 0.04 39 M17 I End 9.4 0.71 0.03 0.01 0.01 0.04 40 j End 9.5 1.86 0.03 0.02 0.02 0.05 53 M18 I End 9.6 1.86 0.03 0.02 0.02 0.05 53 j End 9.8 3.59 0.03 0.04 0.04 0.07 71 M19 I End 10.1 3.59 0.04 0.04 0.04 0.07 72 j End 10.3 4.53 0.04 0.05 0.05 0.08 83 M20 I End 10.6 4.53 0.04 0.05 0.05 0.08 84 j End 10.9 4.73 0.04 0.05 0.05 0.09 87 M21 I End 11.1 4.73 0.04 0.05 0.05 0.09 88 j End 11.6 1.64 0.04 0.02 0.02 0.06 57 M22 I End 12.0 1.64 0.04 0.02 0.02 0.06 59 j End 12.7 3.63 0.04 0.04 0.04 0.01 6

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57 Figure 6 . 32 . The location of truck 2. Figure 6 . 33 . Axial force (kips) due to to load combination of daed load and live load for truck in position 2.

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58 Figure 6 . 34 . Moment (kips ft) due to to load combination of daed load and live load for truck in position 2. 6.4.1.3 Load combination with truck in position 3. Description: (Self Weight + Fill+ Truck 3 + Pressure 1+Pressure2 ) Table 6 . 12 . Total stress due to load combination with truck in position 3. Member Label Section Axial[ k ] Moment [k ft] Axial ( ksi ) Top Bending ( ksi ) Bot Bending ( ksi ) Total Stress ( ksi) Total Stress ( psi ) M1 I End 15.2 0.63 0.05 0.01 0.01 0.06 59 j End 14.5 3.35 0.05 0.03 0.03 0.09 85 M2 I End 14.0 3.35 0.05 0.03 0.03 0.08 83 j End 12.8 4.26 0.04 0.04 0.04 0.09 89 M3 I End 12.2 4.26 0.04 0.04 0.04 0.09 87 j End 11.5 2.58 0.04 0.03 0.03 0.07 67 M4 I End 11.1 2.58 0.04 0.03 0.03 0.07 65 j End 10.6 0.75 0.04 0.01 0.01 0.04 44 M5 I End 10.1 0.75 0.04 0.01 0.01 0.04 43 j End 9.8 1.35 0.03 0.01 0.01 0.02 20 M6 I End 9.7 1.35 0.03 0.01 0.01 0.02 20 j End 9.4 2.35 0.03 0.02 0.02 0.01 8 M7 I End 9.2 2.35 0.03 0.02 0.02 0.01 7

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59 j End 9.0 3.64 0.03 0.04 0.04 0.01 7 M8 I End 9.0 3.64 0.03 0.04 0.04 0.01 7 j End 9.0 4.31 0.03 0.04 0.04 0.01 14 M9 I End 9.0 4.31 0.03 0.04 0.04 0.01 14 j End 9.0 4.59 0.03 0.05 0.05 0.02 17 M10 I End 9.0 4.59 0.03 0.05 0.05 0.02 16 j End 9.0 4.51 0.03 0.05 0.05 0.02 16 M11 I End 9.1 4.51 0.03 0.05 0.05 0.02 15 j End 9.1 4.10 0.03 0.04 0.04 0.01 11 M12 I End 9.1 4.10 0.03 0.04 0.04 0.01 11 j End 9.1 3.77 0.03 0.04 0.04 0.01 7 M13 I End 9.1 3.77 0.03 0.04 0.04 0.01 8 j End 9.1 3.09 0.03 0.03 0.03 0.00 1 M14 I End 9.1 3.09 0.03 0.03 0.03 0.00 1 j End 9.1 2.07 0.03 0.02 0.02 0.01 10 M15 I End 9.1 2.07 0.03 0.02 0.02 0.01 10 j End 9.1 0.65 0.03 0.01 0.01 0.02 25 M16 I End 9.2 0.65 0.03 0.01 0.01 0.03 25 j End 9.2 1.15 0.03 0.01 0.01 0.04 44 M17 I End 9.5 1.15 0.03 0.01 0.01 0.04 45 j End 9.6 2.15 0.03 0.02 0.02 0.06 56 M18 I End 9.7 2.15 0.03 0.02 0.02 0.06 56 j End 9.8 3.72 0.03 0.04 0.04 0.07 73 M19 I End 10.1 3.72 0.04 0.04 0.04 0.07 74 j End 10.3 4.48 0.04 0.05 0.05 0.08 83 M20 I End 10.6 4.48 0.04 0.05 0.05 0.08 84 j End 10.9 4.49 0.04 0.05 0.05 0.08 85 M21 I End 11.1 4.49 0.04 0.05 0.05 0.09 85 j End 11.6 1.14 0.04 0.01 0.01 0.05 52 M22 I End 11.9 1.14 0.04 0.01 0.01 0.05 53 j End 12.6 4.46 0.04 0.05 0.05 0.00 3

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60 Figure 6 . 35 . The location of truck 3. Figure 6 . 36 . Axial force (kips) due to to load combination of daed load and live load for truck in position 3.

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61 Figure 6 . 37 . Moment (kips ft) due to to load combination of daed load and live load for truck in position 3. 6.4.1.4 Load combination with truck in position 4. Description: (Self Weight + Fill+ Truck 3 + Pressure 1+Pressure2 ) Table 6 . 13 . Final results due to load combination with truck in position 4. Member Label Section Axial[ k ] Moment [k ft] Axial (ksi ) Top Bending ( ksi ) Bot Bending ( ksi ) Total Stress ( ksi ) Total Stress (psi ) M1 I End 13.4 0.86 0.05 0.01 0.01 0.04 37 j End 12.3 2.23 0.04 0.02 0.02 0.07 66 M2 I End 11.8 2.23 0.04 0.02 0.02 0.06 64 j End 11.0 3.91 0.04 0.04 0.04 0.08 79 M3 I End 10.6 3.91 0.04 0.04 0.04 0.08 78 j End 10.1 3.01 0.04 0.03 0.03 0.07 67 M4 I End 9.8 3.01 0.03 0.03 0.03 0.07 65 j End 9.4 1.79 0.03 0.02 0.02 0.05 51 M5 I End 9.1 1.79 0.03 0.02 0.02 0.05 50 j End 8.9 0.14 0.03 0.00 0.00 0.03 32 M6 I End 8.8 0.14 0.03 0.00 0.00 0.03 32 j End 8.6 0.72 0.03 0.01 0.01 0.02 22 M7 I End 8.4 0.72 0.03 0.01 0.01 0.02 22 j End 8.3 2.21 0.03 0.02 0.02 0.01 6

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62 M8 I End 8.3 2.21 0.03 0.02 0.02 0.01 6 j End 8.3 3.29 0.03 0.03 0.03 0.01 6 M9 I End 8.3 3.29 0.03 0.03 0.03 0.01 5 j End 8.3 3.95 0.03 0.04 0.04 0.01 12 M10 I End 8.3 3.95 0.03 0.04 0.04 0.01 12 j End 8.3 4.23 0.03 0.04 0.04 0.02 15 M11 I End 8.4 4.23 0.03 0.04 0.04 0.01 15 j End 8.4 4.13 0.03 0.04 0.04 0.01 14 M12 I End 8.4 4.13 0.03 0.04 0.04 0.01 14 j End 8.4 4.05 0.03 0.04 0.04 0.01 13 M13 I End 8.3 4.05 0.03 0.04 0.04 0.01 13 j End 8.3 3.60 0.03 0.04 0.04 0.01 9 M14 I End 8.3 3.60 0.03 0.04 0.04 0.01 9 j End 8.3 2.77 0.03 0.03 0.03 0.00 0 M15 I End 8.3 2.77 0.03 0.03 0.03 0.00 0 j End 8.4 1.52 0.03 0.02 0.02 0.01 13 M16 I End 8.4 1.52 0.03 0.02 0.02 0.01 13 j End 8.4 0.15 0.03 0.00 0.00 0.03 31 M17 I End 8.7 0.15 0.03 0.00 0.00 0.03 32 j End 8.8 1.17 0.03 0.01 0.01 0.04 43 M18 I End 8.9 1.17 0.03 0.01 0.01 0.04 43 j End 9.0 2.78 0.03 0.03 0.03 0.06 60 M19 I End 9.3 2.78 0.03 0.03 0.03 0.06 61 j End 9.5 3.74 0.03 0.04 0.04 0.07 72 M20 I End 9.9 3.74 0.03 0.04 0.04 0.07 73 j End 10.2 4.13 0.04 0.04 0.04 0.08 78 M21 I End 10.4 4.13 0.04 0.04 0.04 0.08 79 j End 10.9 1.62 0.04 0.02 0.02 0.05 55 M22 I End 11.4 1.62 0.04 0.02 0.02 0.06 56 j End 12.0 2.70 0.04 0.03 0.03 0.01 14

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63 Figure 6 . 38 . T he location of truck 4. Figure 6 . 39 . Axial force (kips) due to to load combination of daed load and live load for truck in position 4.

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64 Figure 6 . 40 . Moment (kips ft) due to to load combination of daed load and live load for truck in position 4. 6.4.2 Risa 2D Frame with three hinges . W hen the bending moment in the cross section results in significant net tension, the joints between arch stones will tend to separate. We can approximate this behavior by adding hinges at high tension locations. Risa 2D Frame analysis of the collapse load of the frame leads to high tension in three different locations. Therefore, it is important when solv ing these problems using simple methods to permit joint rotation to simulate stone mas onry joint separations in the real arch. Consequently , hinge forms were assumed in different three positions at joint releases adjacent to N4, N12, and N20 as shown in Figure 6. 4 1 :

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65 Figure 6 . 41 . Model of the arch with hinges in Risa 2D Frame. These hinges leads to reduce moment value and total stress on the members as presents below for all load combination: 6.4.2.1 Load combination with truck in position 1 Table 6 . 14 . Total stress due to load combination with truck in position 1 with three hinges . Member Label Sec tion Axial[k] Moment [k ft] Axial ( ksi ) Top Bending ( ksi ) Bot Bending ( ksi ) Total Stress( ksi ) Total Stress ( psi ) M1 I End 15.21 31.01 0.05 0.32 0.32 0.27 270 j End 14.54 15.05 0.05 0.16 0.16 0.11 106 M2 I End 14.74 15.05 0.05 0.16 0.16 0.11 106 j End 14.26 4.25 0.05 0.04 0.04 0.01 5 M3 I End 15.01 4.25 0.05 0.04 0.04 0.01 8 j End 14.72 0.00 0.05 0.00 0.00 0.05 51 M4 I End 14.86 0.00 0.05 0.00 0.00 0.05 52 j End 14.65 2.26 0.05 0.02 0.02 0.07 74 M5 I End 14.71 2.26 0.05 0.02 0.02 0.07 75 j End 14.57 2.56 0.05 0.03 0.03 0.08 77 M6 I End 14.53 2.56 0.05 0.03 0.03 0.08 77 j End 14.41 3.23 0.05 0.03 0.03 0.08 84

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66 M7 I End 14.36 3.23 0.05 0.03 0.03 0.08 84 j End 14.30 2.12 0.05 0.02 0.02 0.07 72 M8 I End 14.30 2.12 0.05 0.02 0.02 0.07 72 j End 14.25 1.16 0.05 0.01 0.01 0.06 62 M9 I End 14.28 1.16 0.05 0.01 0.01 0.06 62 j End 14.24 0.39 0.05 0.00 0.00 0.05 53 M10 I End 14.28 0.39 0.05 0.00 0.00 0.05 54 j End 14.22 0.06 0.05 0.00 0.00 0.05 49 M11 I End 14.31 0.06 0.05 0.00 0.00 0.05 49 j End 14.28 0.00 0.05 0.00 0.00 0.05 50 M12 I End 14.28 0.00 0.05 0.00 0.00 0.05 50 j End 14.31 0.06 0.05 0.00 0.00 0.05 49 M13 I End 14.22 0.06 0.05 0.00 0.00 0.05 49 j End 14.28 0.39 0.05 0.00 0.00 0.05 54 M14 I End 14.24 0.39 0.05 0.00 0.00 0.05 53 j End 14.28 1.16 0.05 0.01 0.01 0.06 62 M15 I End 14.25 1.16 0.05 0.01 0.01 0.06 62 j End 14.30 2.12 0.05 0.02 0.02 0.07 72 M16 I End 14.30 2.12 0.05 0.02 0.02 0.07 72 j End 14.36 3.23 0.05 0.03 0.03 0.08 84 M17 I End 14.41 3.23 0.05 0.03 0.03 0.08 84 j End 14.53 2.56 0.05 0.03 0.03 0.08 77 M18 I End 14.57 2.56 0.05 0.03 0.03 0.08 77 j End 14.71 2.26 0.05 0.02 0.02 0.07 75 M19 I End 14.65 2.26 0.05 0.02 0.02 0.07 74 j End 14.86 0.00 0.05 0.00 0.00 0.05 52 M20 I End 14.72 0.00 0.05 0.00 0.00 0.05 51 j End 15.01 4.25 0.05 0.04 0.04 0.01 8 M21 I End 14.26 4.25 0.05 0.04 0.04 0.01 5 j End 14.74 15.05 0.05 0.16 0.16 0.11 106 M22 I End 14.54 15.05 0.05 0.16 0.16 0.11 106 j End 15.21 31.01 0.05 0.32 0.32 0.27 270

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67 Figure 6 . 42 . Axial force (kips) due to l oad combination of dead load and live load for truck in position 1 with three hinges . Figure 6 . 43 . Moment (kips ft) due to load combination of dead load and live load for truck in position 1 with three hinges .

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68 6.4.2.2 Load combination with truck in position 2 Table 6 . 15 . Total stress due to load combination with truck in position 2 with three hinges . Member Label Section Axial[ k ] Moment [k ft] Axial ( ksi ) Top Bending ( ksi ) Bot Bending ( ksi ) Total Stress( ksi ) Total Stress ( psi ) M1 I End 15.15 26.84 0.05 0.28 0.28 0.23 227 j End 14.48 12.70 0.05 0.13 0.13 0.08 82 M2 I End 14.58 12.70 0.05 0.13 0.13 0.08 82 j End 14.09 3.31 0.05 0.03 0.03 0.01 15 M3 I End 14.65 3.31 0.05 0.03 0.03 0.02 16 j End 14.37 0.00 0.05 0.00 0.00 0.05 50 M4 I End 14.41 0.00 0.05 0.00 0.00 0.05 50 j End 14.19 1.48 0.05 0.02 0.02 0.06 65 M5 I End 14.16 1.48 0.05 0.02 0.02 0.06 65 j End 14.02 1.18 0.05 0.01 0.01 0.06 61 M6 I End 13.96 1.18 0.05 0.01 0.01 0.06 61 j End 13.77 1.32 0.05 0.01 0.01 0.06 62 M7 I End 13.67 1.32 0.05 0.01 0.01 0.06 61 j End 13.54 0.08 0.05 0.00 0.00 0.05 48 M8 I End 13.55 0.08 0.05 0.00 0.00 0.05 48 j End 13.44 0.67 0.05 0.01 0.01 0.04 40 M9 I End 13.49 0.67 0.05 0.01 0.01 0.04 40 j End 13.41 0.90 0.05 0.01 0.01 0.04 37 M10 I End 13.48 0.90 0.05 0.01 0.01 0.04 37 j End 13.43 0.66 0.05 0.01 0.01 0.04 40 M11 I End 13.54 0.66 0.05 0.01 0.01 0.04 40 j End 13.52 0.00 0.05 0.00 0.00 0.05 47 M12 I End 13.56 0.00 0.05 0.00 0.00 0.05 47 j End 13.58 0.26 0.05 0.00 0.00 0.05 50 M13 I End 13.50 0.26 0.05 0.00 0.00 0.05 50 j End 13.53 0.67 0.05 0.01 0.01 0.05 54 M14 I End 13.47 0.67 0.05 0.01 0.01 0.05 54 j End 13.51 1.24 0.05 0.01 0.01 0.06 60 M15 I End 13.47 1.24 0.05 0.01 0.01 0.06 60 j End 13.53 2.03 0.05 0.02 0.02 0.07 68 M16 I End 13.52 2.03 0.05 0.02 0.02 0.07 68 j End 13.59 3.01 0.05 0.03 0.03 0.08 78 M17 I End 13.62 3.01 0.05 0.03 0.03 0.08 79 j End 13.74 2.34 0.05 0.02 0.02 0.07 72

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69 M18 I End 13.78 2.34 0.05 0.02 0.02 0.07 72 j End 13.92 2.06 0.05 0.02 0.02 0.07 70 M19 I End 13.88 2.06 0.05 0.02 0.02 0.07 70 j End 14.09 0.00 0.05 0.00 0.00 0.05 49 M20 I End 13.98 0.00 0.05 0.00 0.00 0.05 49 j End 14.27 3.88 0.05 0.04 0.04 0.01 9 M21 I End 13.61 3.88 0.05 0.04 0.04 0.01 7 j End 14.10 13.85 0.05 0.14 0.14 0.10 95 M22 I End 13.96 13.85 0.05 0.14 0.14 0.10 96 j End 14.63 28.58 0.05 0.30 0.30 0.25 247 Figure 6 . 44 . Axial force (kips) due to load combination of dead load and live load for truck in position 2 with three hinges .

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70 Figure 6 . 45 . Moment (kips ft) due to load combination of dead load and live load for truck in position 2 with three hinges . 6.4.2.3 Load combination with truck in position 3 Table 6 . 16 . Total stress due to load combination with truck in position 3 with three hinges . Member Label Sec tion Axial[k] Moment [k ft] Axial ( ksi ) Top Bending ( ksi ) Bot Bending ( ksi ) Total Stress ( ksi) Total Stress ( psi ) M1 I End 16.87 17.75 0.06 0.18 0.18 0.13 126 j End 16.20 7.67 0.06 0.08 0.08 0.02 24 M2 I End 16.05 7.67 0.06 0.08 0.08 0.02 24 j End 14.83 1.43 0.05 0.01 0.01 0.04 37 M3 I End 14.99 1.43 0.05 0.01 0.01 0.04 37 j End 14.27 0.00 0.05 0.00 0.00 0.05 50 M4 I End 14.14 0.00 0.05 0.00 0.00 0.05 49 j End 13.61 0.42 0.05 0.00 0.00 0.05 52 M5 I End 13.48 0.42 0.05 0.00 0.00 0.05 51 j End 13.12 0.21 0.05 0.00 0.00 0.04 43 M6 I End 13.07 0.21 0.05 0.00 0.00 0.04 43 j End 12.75 0.11 0.04 0.00 0.00 0.05 45 M7 I End 12.71 0.11 0.04 0.00 0.00 0.05 45 j End 12.55 0.55 0.04 0.01 0.01 0.04 38

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71 M8 I End 12.58 0.55 0.04 0.01 0.01 0.04 38 j End 12.53 0.75 0.04 0.01 0.01 0.04 36 M9 I End 12.59 0.75 0.04 0.01 0.01 0.04 36 j End 12.55 0.69 0.04 0.01 0.01 0.04 36 M10 I End 12.63 0.69 0.04 0.01 0.01 0.04 37 j End 12.60 0.44 0.04 0.00 0.00 0.04 39 M11 I End 12.71 0.44 0.04 0.00 0.00 0.04 40 j End 12.69 0.00 0.04 0.00 0.00 0.04 44 M12 I End 12.71 0.00 0.04 0.00 0.00 0.04 44 j End 12.73 0.07 0.04 0.00 0.00 0.04 45 M13 I End 12.64 0.07 0.04 0.00 0.00 0.04 45 j End 12.67 0.33 0.04 0.00 0.00 0.05 47 M14 I End 12.61 0.33 0.04 0.00 0.00 0.05 47 j End 12.65 0.78 0.04 0.01 0.01 0.05 52 M15 I End 12.61 0.78 0.04 0.01 0.01 0.05 52 j End 12.66 1.49 0.04 0.02 0.02 0.06 59 M16 I End 12.65 1.49 0.04 0.02 0.02 0.06 59 j End 12.72 2.42 0.04 0.03 0.03 0.07 69 M17 I End 12.75 2.42 0.04 0.03 0.03 0.07 69 j End 12.88 1.86 0.04 0.02 0.02 0.06 64 M18 I End 12.92 1.86 0.04 0.02 0.02 0.06 64 j End 13.07 1.73 0.05 0.02 0.02 0.06 63 M19 I End 13.05 1.73 0.05 0.02 0.02 0.06 63 j End 13.26 0.00 0.05 0.00 0.00 0.05 46 M20 I End 13.21 0.00 0.05 0.00 0.00 0.05 46 j End 13.49 3.35 0.05 0.03 0.03 0.01 12 M21 I End 12.96 3.35 0.05 0.03 0.03 0.01 10 j End 13.45 12.27 0.05 0.13 0.13 0.08 81 M22 I End 13.40 12.27 0.05 0.13 0.13 0.08 81 j End 14.06 25.47 0.05 0.27 0.27 0.22 216

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72 Figure 6 . 46 . Axial force (kips) due to load combination of dead load and live load for truck in position 3 with three hinges . Figure 6 . 47 . Moment (kips ft) due to load combination of dead load and live load for truck in position 3 with three hinges.

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73 6.4.2.4 Load combination with truck in position 4 Table 6 . 17 . Total stress due to load combination with truck in position 4 with three hinges. Member Label Sec tion Axial[k] Moment [k ft] Axial (ksi ) Top Bending (ksi ) Bot Bending (ksi ) Total Stress (ksi ) Total Stress (psi ) M1 I End 14.92 19.64 0.05 0.20 0.20 0.15 153 j End 13.84 9.23 0.05 0.10 0.10 0.05 48 M2 I End 13.74 9.23 0.05 0.10 0.10 0.05 48 j End 12.96 2.24 0.04 0.02 0.02 0.02 22 M3 I End 13.25 2.24 0.05 0.02 0.02 0.02 23 j End 12.78 0.00 0.04 0.00 0.00 0.04 44 M4 I End 12.73 0.00 0.04 0.00 0.00 0.04 44 j End 12.39 1.06 0.04 0.01 0.01 0.05 54 M5 I End 12.34 1.06 0.04 0.01 0.01 0.05 54 j End 12.11 0.93 0.04 0.01 0.01 0.05 52 M6 I End 12.06 0.93 0.04 0.01 0.01 0.05 52 j End 11.90 1.43 0.04 0.01 0.01 0.06 56 M7 I End 11.87 1.43 0.04 0.01 0.01 0.06 56 j End 11.80 0.64 0.04 0.01 0.01 0.05 48 M8 I End 11.82 0.64 0.04 0.01 0.01 0.05 48 j End 11.77 0.09 0.04 0.00 0.00 0.04 42 M9 I End 11.81 0.09 0.04 0.00 0.00 0.04 42 j End 11.77 0.17 0.04 0.00 0.00 0.04 39 M10 I End 11.84 0.17 0.04 0.00 0.00 0.04 39 j End 11.82 0.19 0.04 0.00 0.00 0.04 39 M11 I End 11.91 0.19 0.04 0.00 0.00 0.04 39 j End 11.90 0.00 0.04 0.00 0.00 0.04 41 M12 I End 11.90 0.00 0.04 0.00 0.00 0.04 41 j End 11.92 0.11 0.04 0.00 0.00 0.04 40 M13 I End 11.82 0.11 0.04 0.00 0.00 0.04 40 j End 11.85 0.00 0.04 0.00 0.00 0.04 41 M14 I End 11.78 0.00 0.04 0.00 0.00 0.04 41 j End 11.82 0.34 0.04 0.00 0.00 0.04 45 M15 I End 11.78 0.34 0.04 0.00 0.00 0.04 44 j End 11.83 0.97 0.04 0.01 0.01 0.05 51 M16 I End 11.82 0.97 0.04 0.01 0.01 0.05 51 j End 11.89 1.85 0.04 0.02 0.02 0.06 61 M17 I End 11.93 1.85 0.04 0.02 0.02 0.06 61 j End 12.05 1.40 0.04 0.01 0.01 0.06 56 M18 I End 12.10 1.40 0.04 0.01 0.01 0.06 57

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74 j End 12.24 1.41 0.04 0.01 0.01 0.06 57 M19 I End 12.26 1.41 0.04 0.01 0.01 0.06 57 j End 12.47 0.00 0.04 0.00 0.00 0.04 43 M20 I End 12.46 0.00 0.04 0.00 0.00 0.04 43 j End 12.75 2.84 0.04 0.03 0.03 0.01 15 M21 I End 12.34 2.84 0.04 0.03 0.03 0.01 13 j End 12.83 10.75 0.04 0.11 0.11 0.07 67 M22 I End 12.85 10.75 0.04 0.11 0.11 0.07 67 j End 13.52 22.49 0.05 0.23 0.23 0.19 187 Figure 6 . 48 . Axial force (kips) due to l oad combination of dead load and live load for truck in position 4 with three hinges .

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75 Figure 6 . 49 . Moment (kips ft) due to load combination of dead load and live load for truck in position 4 with three hinges . 6.5 Analysis U sing Risa 2D F inite Element . This method is based on Finite Element. The arch is model led as a many plates which have the same properties. Also, in this analysis, fixed supports are appl ied on the abutments. Figure 6.5 0 shows the model of the arch using Risa Finite Element. Figure 6 . 50 . Finit element models .

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76 6.5.1 Loads . Table 6 . 18 . Dead load values in Risa FE. Node # Self Wight Load ( k/ft) Fill Load (k/ft) Pressure Load (Right) (k/ft) Pressure Load (Left) (k/ft) N1 0.32 1.17 0.351 N2 0.32 0.92004 0.276012 N3 0.32 0.738 0.2214 N4 0.32 0.63 0.189 N5 0.32 0.5508 0.16524 N6 0.32 0.498 0.1494 N7 0.32 0.45 0.135 N8 0.32 0.4248 0.12744 N9 0.32 0.405 0.1215 N10 0.32 0.39 0.117 N11 0.32 0.38004 0.114012 N12 0.32 0.375 0.1125 N13 0.32 0.38004 0.114012 N14 0.32 0.39 0.117 N15 0.32 0.405 0.1215 N16 0.32 0.4248 0.12744 N17 0.32 0.45 0.135 N18 0.32 0.498 0.1494 N19 0.32 0.5508 0.16524 N20 0.32 0.63 0.189 N21 0.32 0.738 0.2214 N22 0.32 0.92004 0.276012 N23 0.32 1.17 0.351

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77 Figure 6 . 51 . Self weight load Fill load in Risa FE . Figure 6 . 52 . Pressure load on the right side of the arch in Risa FE .

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78 Figure 6 . 53 . Pressure load on the left side of the arch in Risa FE . T he Ely Stone Bridge was applied the dead load and live load with four different load combustion based on four positions. T he description of the position of live load and the result for dead load and live load is shown below: Figure 6 . 54 . Truck 1 position in Risa FE.

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79 Figure 6 . 55 . Truck 2 position in Risa FE. Figure 6 . 56 . Truck 3 position in Risa FE.

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80 Fig ure 6 . 57 . Truck 4 position in Risa FE. Figure 6 . 58 . The stress by dead load in Risa FE.

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81 Figure 6 . 59 . The stress by in Risa FE. Figure 6 . 60 . The stress by in Risa FE.

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82 Figure 6 . 61 . The stress by in Risa FE. Figure 6 . 62 . The stress by in Risa FE.

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83 CHAPTER VII 7 DISCUSSION 7.1 Introduction to Comparison of D ifferent C alculation M ethods There are many empirical and numerical methods that help to determine the failure load with 2D and 3D modell ing. Use of these methods help s us to get more understanding of the behavior of the structure under multiple loads. Using these k inds of methods bring s many, but not all effect s and circumstance s into consideration. However, it is necessary to spen d much time in creating models. 7.1.1 The MEXE and Ring A comparative calculation was done for the same bridge by using two different methods which are the MEXE and Ring methods . A single axle was located over the span and then the ultimat e load calculate was calculated and the results shown in Table 7.1. Table 7 . 1 . Results from MEXE method and Ring. Method Permissible axle load MEXE Method 120 kips Ring 3.2 Method 220 kip s The result from the MEXE Modified method , an empirical method, is based in part on with regard to condition . The MEXE method can be a helpful method to estimate the load capacity for masonry arch bridges. On the other hand, the result from Ring is based on the attributes of the bridge such as fill angle of internal friction, f riction coefficient between stones, and unit weight for the material. S ome of parameter s were assumed because it is impossible to measure them in an existing bridge, i.e.

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84 friction coefficients between the stones of the arch barrel and compressive strength of the masonry. From the comparison results on the T able above, the MEXE method gives a permissible axle load of 120 kip. However, Ring analysis ga ve a permissible a xle load 220 kip for a single axle. Hence, the Ring method gives a permissible axle load about 2.0 times th e p ermissible axle load from the MEXE method. The big gest reason for this difference in the load factor value is that Ring software take s into consideration arch failure and already applied hinges on those locations that have a mechanism failure which can lead to collapse, whereas MEXE is a more simplistic and inherently conservative approach. 7.1.2 Risa 2D F rame and Ring . Different comparative calculation s were done here between Risa 2D Frame and Ring. In this comparison, the design load factor for the Ely Stone Bridge using the Ring method was applied as a live load on the middle of the strip of arch barrel using the Risa 2D Fram e and Ring software as shown in the Figure below : Figure 7 . 1 . Model of the Ely Stone Bridge in Risa 2D Frame.

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85 Figure 7 . 2 . Model of the Ely Stone Bridge in Ring. The results for the Ely Stone Bridge from two methods match better than those for the MEXE and Ring methods. Where the analysi s for both methods gives the same critical locations, which have tension area as seen in the axial force diagram and the moment diagram Figures below: Figure 7 . 3 . Axial force (kips) diagram load, in Risa 2D.

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86 Figure 7 . 4 . Moment diagram by the critical design load in Risa 2D. Figure 7 . 5 . Failure mechanism for the The Ely Stone Bridge from RING2.0.

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87 Figure 7 . 6 . Axial force diagram by the critical design load in Ring. Figure 7 . 7 . Moment diagram by the criti cal design load in Ring .

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88 From the above results, there are three high mom ent areas. At these high moment areas, joint releases adjacent to N4, N12, and N20, have been introduced to simulate hinges. This is approximately model the effect of joint separations between stones in the arch. Figure 7 . 8 . Model of the Ely Stone Bridge under critical l oad in Risa 2D Frame with three hinges. Figure 7 . 9 . Axial force (kips) diagram by the critical de sign load in Risa 2D with three hinges.

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89 Figure 7 . 10 . Moment (kips ft) diagram by the critical de sign load in Risa 2D with three hinges. From the previous result, we see that the arch in Risa 2D Frame analysis needs to have three hinges , similar to the Ring analysis, to reduce the tension area. However, Risa 2D Frame still shows high tension near the fixed ends . 7.2 Conclusion This research was done by using four different methods to analyze and assess the Ely S tone Bridge. Two of the methods, the MEXE method and Ring are used to determine the load capacity of the Ely Bridge. These two methods are simple and quick to learn. While Ring is a complex software, it is very fast running analysis. The results show that the MEXE and Ring method s give different results for the same bridge under the same load . MEXE leads to a more conservative re sult. Risa 2D Frame and Risa Finite Elements are programs which give us the stress that is created under applied loads, they are difficult to compare with MEXE and Ring because their l oads. The main

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90 reason which makes the comparison difficult is that Ring sof tware can apply more than three hinges on the structure. However, Risa cannot apply more than three hinges . 7.3 Recommendations for Further Research It is possible to create a three d imensional model for a linearly elastic analysis of the Ely Stone Bridge. It would probably lead to a more comprehensive analysis leading to increase of the load factor of the bridge. However, this kind of modeling presents more analytical challenges becau se hing e formation in three dimensions, making the analysis non linear, would be necessary. Future analysis would benefit from experimental work.

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91 REFERENCES Ålenius , M. (2003). Finite e lement m odelling of c omposite b ridge s tability . Bader, M. (September 2016). Discussion with UCD team. Beyer, L. (2012). Arched Bridges History and Analysis . University of New Hampshire . Bögöly, G. (2012 ). Diagnosti Conference of Junior Researchers in Civil Engineering, 28 33 . Bjurstrom. H., Lasell. J. (2009). Capacity assessment of the single span arch bridge with backfill. A case study of Glomman Bridge. Citto , C., Woodham , D.B. (2015). Evaluating existing and historic stone arch bridge. Structure Forensics . Citto , C. , Woodham , D.B. (2016). Modeling unknown variables and degradation effects in the assessment of stone arch bridges . Brick and Block Masonry , 1055 1062. Citto , C., Woodham Evaluation and r ating of m asonry a rch b ridges Structures Congress , 528 538 . Drysdale, R ., Hamid, A. (2008). Corbelled arches . Masonry Structure Behavior and Design , 14. Harvey Jr. D. W ., Schuller, M. P. Structural Performance of Masonry . Holmstrom, K. (2010). On engineering methods for assessment of load capacity of stone arch bridges . Chalmers University of Technology . Jackson, D.C. (1988). National Trust for Historic Preservation. A National Trust Guide, Great American Bridges and Dams . Limitstate. (2016). Version 3.2.a , R apid masonry arch analysis software . Martinez, J. , Revilla, M., Torre , A. (2001). Critical thickness criteria on stone arch bridges with low rise/span ratio and current traffic loads . Historical Constructions . 609 616. Monaco , M,. Frunzio . G., Gesualdo 3D F.E.M. a nalysis of a Roman arch bridge. Historical Constructi 591 598 . McClain, Set of Drawings . (2013). Grading H.M.A. Pavement on Stone Bridge over Wet Creek . for Iowa Dept. of Transportation , Highway Division, Jones County.

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92 National Register of Historic Places . (1979r) rvice, US Dept. of the Interior. Netolicky, L. (Sep 2016) . " Study Begins on Ely Stone Bridge .". Journal Eureka. N.P RISA (2013). RISA 2D , Ver.12.0, RISA Technologies, Foothill Ranch, CA. MEXE Method. Robinson BEng , J. (2000). Analysis and a ssessment of m asonry a rch b ridges . University of Edinburgh . Structure Inventory and Appraisal, Bridge ID: C 2649 , for Iowa DOT, 2013. Structure Inventory and Appraisal, Bridge ID: C 2649 , for Iowa DOT, 2015. Steinmetz, Douglas J. ( Dec 2016). Preservation Commission . The highways agency, (2001). The Assessment of Highway Bridges and S tructures . Three Hinged Arch , Version 2 CE IIT . Cables and Arches . Yang, Y. (1991). Progressive failure analysis of masonry arch bridges. Zhang, Y. (2015). Advanced nonlinear analysis of masonry arch bridges.

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93 APPENDIX A Field Visitation On Sept. 8, 9, and 10, 2016, faculty, students, and staff from the Civil Engineering Department of the University of Colorado Denver (UCD) went to Iowa to visit the Ely Sto ne Bridge to inspect the bridge, to s urvey data, collect information, and get more docu mentation about the bridge. A load test was also conducted . They also met with public officials. The UCD team acquired historic photographs that date from 1919. Also, the community members contributed some other historic photographs of the Ely Stone Bridg e. From the data collected and these photos, the UCD team assessed the condition of the bridge over the past years ( C ondition A ssessment R eport, 2016). Numbering system. There is a numbering system tha t has been assumed for the Ely Stone B ridge for spans, supporting abutments and piers from south to north. However, the following assumptions have been made by the E ly Stone Bridge team to identify and organize the work on the Ely Stone Bridge as following: T The upstream side located on the west and the downstream side is on the east. The supporting abutments and piers are also numbered from south to north, thus Abutment 1 (A1) is at the south, followed by Pier 2 (P2), Pier 3 (P3) and Abutment 4 (A4) at the north.

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94 Figure A.1 . Side view of the Ely Stone Bridge at east side represent the numbering system. General condition assessment. The UCD team surveyed the site to determine the geometry and dimensions of the bridge, including span length, arch height , pier dimensions, roadway width and depth etc. So, the modeling on the computer will be based on information that was derived from this survey. According to some photos received by the UCD team t here has been some rebuilding work completed on parts of the Ely Stone B ridge, such as its stone facing, pavement and steel guard rails during 2003 and 2004 as shown in F igures A.2. and A.3. below:

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95 Figure A.2 . Condition of the bridge before rebuilding work in 2002. Figure A.3 . Condition of the bridge a fter rebuilding work in 2003 . Mortar condition assessment. The original mortar which was used in 1893 appears to be weak and disintegrating to the point where it can be extracted by a small tool or even by hand as shown in Figure A.4 .

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96 T he old mortar may be sand lime or a type of soft cement. Other more modern methods used to stabilize th e older deteriorating mortar have been unsuccessful. Figure A.4. Represent the condition of mortar. When the UCD research team visited the site of the bridge, they witnessed a variety of properties of mortars such as different colors and different textures. This means the original mortar has been partially replaced. After this visit, Architect David Arbogast ha s done some investig ati ng and discover ed that the original mortar is a sand lime mortar which is softer that mortars containing Portland cement (Arbogast , 2016 ) . T he NRHP nomination reports and investigations shows that there are some maintenance activities done i n 1933, 195 5, 2003, 2010, and 2013. So, this is a big reason for the discovery of v aryin g shades of grey color and hardness. In the F igure A.5. below , it is clear to see where the new mortar is located near of the surface of stones and the possibly the original mortar, which is located deeper and ha s dark color .

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97 Figure A.5 . Represent the condition of mortar (old and new mortar). F igure A.6. below illustrates the location of the spalling. The combination of hard and soft mortars can cause spalling, which is los s of stone or other material on the face of a mas onry unit through flaking. I t is a phenomenon that has occurred in several locations on the bridge, between the stones and a redistribution of the load paths. Figure A.6 . Example of where spalling of the stone surface has occurred.

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98 Finally, based on all information from Preservation Brief 2 a nd the report of (Mr. Steinmetz, 2016), the UCD team summarize d that the Ely Stone Bridge faced spalling issue s which can be create d from differ ent k inds of mortar (soft and hard) , the team predicts the spalling could continue . Masonry facades assessment. B ased on visual observations , cracking was observed at the arch. T hey are different from one to another in length and width and most of the crac ks were found in the upper portion of the arch as seen in Figure A.7. Figure A.7 . Cracking on the bridge, west side, 2016. In addition to the cracking issue , there is an other problem which is missing facing stone. On the west side of the Ely Stone Bridge, there are two missing stone areas that create a big hole in the facing vi ew as shown on F igures A.8. and A.9. below. So, it is clear that the east side facing is in better situation than the west side.

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99 Figure A.8. Missing facing stone on the brid ge, west side, 2016. Figure A.9. Missing facing stone on the bridge, west side, 2016.

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100 Limestones The shape of limestones on the arches are similar except on the bottom of the arch course. Unlike other typical arch structure s all of the stones in the arch are close to being rec tangular cut as seen in Figure A.10. Figure A.10 . The shape of limestones on the a rch . T he top of the pavement and top of the stone bridge are at the sa me level as at the middle of the bridge. However, on the ends of the bridge (north and south), the top pavement is higher than the top of the stonework.

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101 Figure A.11 . The top of bridge on the middle. Figure A.12 . The top of bridge as viewed from both ends.

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102 Protective Concrete Collars T he Ely Stone Bridge has reinforced concrete to protect its pier foundation s . The main reason to use reinforced concrete is to protect the stone which, was originally designed to protect the bridge from raging water a nd ice. The Ely Stone Br idge has suffered impairment because it was not built with modern material such as steel or concrete. Figure A.13 . The shape of th e concrete protection at the Ely Stone Bridge.

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103 Figure A.14 . The concrete pr otection at west side of the Ely Stone Bridge. Conclusion The bridge continues to carry cars, trucks, and other vehicles since 1893, but with some maintenance issues over the years.

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104 APPENDIX B S preadsheet for MEXE M ethods The Modified MEXE method. L: The span (m) r c :The rise of the arch at the crown (m) r q :The rise of the arch at the quarter points (m) d : The thickness of the arch barrel to the keystone (m) h : The depth of the fill (m) ft m L 19.82 6.0 r c 5.6 1.7 r q 3 0.9 d 2 0.6 h 2.1 5 0.7 PAL : The provisional axle loading PAL 114.2624789 PAL 70 tons Span/Rise Factor (Fsr) Material Factor (Fm) Joint Factor (Fi) Fsr = 3.539285714 Fsr = 1 Fp= 1.5 Fp= 1.0 Fb= 1 Ff= 0.7 Fm 1.0

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105 Fw= 0.8 Fd= 1 Fmo= 1 Fj 0.8 Condition Factor (Fe) Fe= 1 Modified axle load (Wm) Wm 54 Tons 120 Kips

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106 APPENDIX C Report from RING 3.2 A nalysis Analysis for one span of the Ely Stone Bridge , Single Axle (1kN)

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107

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108

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109 Analysis for three spans of the Ely Stone Bridge, Single Axle (1kN)

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